Advances in Operator Theory Articles (Project Euclid)
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The latest articles from Advances in Operator Theory on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Mon, 04 Dec 2017 14:36 ESTMon, 04 Dec 2017 14:36 ESThttps://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Square inequality and strong order relation
https://projecteuclid.org/euclid.aot/1512416206
<strong>Tsuyoshi Ando</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 1, Number 1, 1--7.</p><p><strong>Abstract:</strong><br/>
It is well-known that for Hilbert space linear operators $0 \leq A$ and $0 \leq C$, inequality $C \leq A$ does not imply $C^2 \leq A^2.$ We introduce a strong order relation $0 \leq B \lll A$, which guarantees that $C^2 \leq B^{1/2}AB^{1/2}$ text for all $0 \leq C \leq B,$ and that $C^2 \leq A^2$ when $B$ commutes with $A$. Connections of this approach with the arithmetic-geometric mean inequality of Bhatia-Kittaneh as well as the Kantorovich constant of $A$ are mentioned.
</p>projecteuclid.org/euclid.aot/1512416206_20171204143649Mon, 04 Dec 2017 14:36 ESTSome lower bounds for the numerical radius of Hilbert space operatorshttps://projecteuclid.org/euclid.aot/1512431558<strong>Ali Zamani</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 2, 98--107.</p><p><strong>Abstract:</strong><br/>
We show that if $T$ is a bounded linear operator on a complex Hilbert space, then $$\frac{1}{2} ||T|| \leq \sqrt {{\frac{w^2(T)}{2}} + \frac{w(T)}{2} \sqrt{w^2(T) - c^2(T)}} \leq w(T),$$ where $w(\cdot)$ and $c(\cdot)$ are the numerical radius and the Crawford number, respectively. We then apply it to prove that for each $t \in [0, \frac {1}{2})$ and natural number $k$, $$\frac {(1 + 2t)^{\frac{1}{2k}}}{{2}^{\frac{1}{k}}}m(T)\leq w(T),$$ where $m(T)$ denotes the minimum modulus of $T$. Some other related results are also presented.
</p>projecteuclid.org/euclid.aot/1512431558_20171204185243Mon, 04 Dec 2017 18:52 ESTOn maps compressing the numerical range between $C^*$-algebrashttps://projecteuclid.org/euclid.aot/1512431559<strong>Aschwag Fahad Albideewi</strong>, <strong>Mohamed Mabruk</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 2, 108--113.</p><p><strong>Abstract:</strong><br/>
In this paper, we deal with the problem of characterizing linear maps compressing the numerical range. A counterexample is given to show that such a map need not be a Jordan $*$-homomorphism in general even if the $C^*$-algebras are commutative. Next, under an auxiliary condition we show that such a map is a Jordan $*$-homomorphism.
</p>projecteuclid.org/euclid.aot/1512431559_20171204185243Mon, 04 Dec 2017 18:52 ESTNormalized tight vs. general frames in sampling problemshttps://projecteuclid.org/euclid.aot/1512431560<strong>Tomaž Košir</strong>, <strong>Matjaž Omladič</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 2, 114--125.</p><p><strong>Abstract:</strong><br/>
We present a new approach to sampling theory using the operator theory framework. We use a replacement operator and replace general frames of the sampling and reconstruction subspaces by normalized tight frames. The replacement can be done in a numerically stable and efficient way. The approach enables us to unify the standard consistent reconstruction results with the results for quasiconsistent reconstruction. Our approach naturally generalizes to consistent and quasiconsistent reconstructions from several samples. Not only we can handle sampling problems in a more efficient way, we also answer questions that seem to be open so far.
</p>projecteuclid.org/euclid.aot/1512431560_20171204185243Mon, 04 Dec 2017 18:52 ESTReproducing pairs of measurable functions and partial inner product spaceshttps://projecteuclid.org/euclid.aot/1512431561<strong>Jean-Pierre Antoine</strong>, <strong>Camillo Trapani</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 2, 126--146.</p><p><strong>Abstract:</strong><br/>
We continue the analysis of reproducing pairs of weakly measurable functions, which generalize continuous frames. More precisely, we examine the case where the defining measurable functions take their values in a partial inner product space (PIP spaces). Several examples, both discrete and continuous, are presented.
</p>projecteuclid.org/euclid.aot/1512431561_20171204185243Mon, 04 Dec 2017 18:52 ESTSome results about fixed points in the complete metric space of zero at infinity varieties and complete convex metric space of varietieshttps://projecteuclid.org/euclid.aot/1512431562<strong>Ghorban Khalilzadeh Ranjbar</strong>, <strong>Tooraj Amiri</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 2, 147--161.</p><p><strong>Abstract:</strong><br/>
This paper aims to study fixed points in the complete metric space of varieties which are zero at infinity as a subspace of the complete metric space of all varieties. Also, the convex structure of the complete metric space of all varieties will be introduced.
</p>projecteuclid.org/euclid.aot/1512431562_20171204185243Mon, 04 Dec 2017 18:52 ESTDirect estimates of certain Miheşan-Durrmeyer type operatorshttps://projecteuclid.org/euclid.aot/1512431563<strong>Arun Kajla</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 2, 162--178.</p><p><strong>Abstract:</strong><br/>
In this note we consider a Durrmeyer type operator having the basis functions in summation and integration due to Mihecşan [Creative Math. Inf. 17 (2008), 466-472.] and Pvǎltvǎnea [Carpathian J. Math. 24 (2008), no. 3, 378-385.] that preserve the linear functions. We present a Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. In the last section of the paper, we obtain the rate of approximation for absolutely continuous functions having a derivative equivalent with a function of bounded variation.
</p>projecteuclid.org/euclid.aot/1512431563_20171204185243Mon, 04 Dec 2017 18:52 ESTOn spectral synthesis in several variableshttps://projecteuclid.org/euclid.aot/1512431564<strong>László Székelyhidi</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 2, 179--191.</p><p><strong>Abstract:</strong><br/>
In a recent paper we proposed a possible generalization of L. Schwartz's classical spectral synthesis result for continuous functions in several variables. The idea is based on Gelfand pairs and spherical functions while "translation invariance" is replaced by invariance with respect to the action of affine groups. In this paper we describe the function classes which play the role of the exponential monomials in this setting.
</p>projecteuclid.org/euclid.aot/1512431564_20171204185243Mon, 04 Dec 2017 18:52 ESTOn the weak compactness of Weak* Dunford-Pettis operators on Banach latticeshttps://projecteuclid.org/euclid.aot/1512431670<strong>El Fahri Kamal</strong>, <strong>H'michane Jawad</strong>, <strong>El Kaddouri Abdelmonim</strong>, <strong>Aboutafail Moulay Othmane</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 192--200.</p><p><strong>Abstract:</strong><br/>
We characterize Banach lattices on which each positive weak* Dunford-Pettis operator is weakly (resp., M-weakly, resp., order weakly) compact. More precisely, we prove that if $F$ is a Banach lattice with order continuous norm, then each positive weak* Dunford-Pettis operator $T : E \longrightarrow F$ is weakly compact if, and only if, the norm of $E^{\prime}$ is order continuous or $F$ is reflexive. On the other hand, when the Banach lattice $F$ is Dedekind $\sigma$-complete, we show that every positive weak* Dunford-Pettis operator $T: E \longrightarrow F$ is M-weakly compact if, and only if, the norms of $E^{\prime}$ and $F$ are order continuous or $E$ is finite-dimensional.
</p>projecteuclid.org/euclid.aot/1512431670_20171204185436Mon, 04 Dec 2017 18:54 ESTTwo-weight norm inequalities for the higher-order commutators of fractional integral operatorshttps://projecteuclid.org/euclid.aot/1512431671<strong>Caiyin Niu</strong>, <strong>Xiaojin Zhang</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 201--214.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain several sufficient conditions such that the higher-order commutators $I_{\alpha,b}^m$ generated by $I_\alpha$ and $b\in \textrm{BMO}(\mathbb{R}^n)$ is bounded from $L^p(v)$ to $L^q(u)$, where $\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}$ and $0 \lt \alpha \lt n$.
</p>projecteuclid.org/euclid.aot/1512431671_20171204185436Mon, 04 Dec 2017 18:54 ESTProperties of $J$-fusion frames in Krein spaceshttps://projecteuclid.org/euclid.aot/1512431672<strong>Shibashis Karmakar</strong>, <strong>Sk. Monowar Hossein</strong>, <strong>Kallol Paul</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 215--227.</p><p><strong>Abstract:</strong><br/>
In this article we introduce the notion of $J$-Parseval fusion frames in a Krein space $\mathbb{K}$ and characterize 1-uniform $J$-Parseval fusion frames with $\zeta=\sqrt{2}$. We provide some results regarding construction of new $J$-tight fusion frame from given $J$-tight fusion frames. We also characterize an uniformly $J$-definite subspace of a Krein space $\mathbb{K}$ in terms of $J$-fusion frame. Finally we generalize the fundamental identity of Hilbert space frames in the setting of Krein spaces.
</p>projecteuclid.org/euclid.aot/1512431672_20171204185436Mon, 04 Dec 2017 18:54 ESTOn the behavior at infinity of certain integral operator with positive kernelhttps://projecteuclid.org/euclid.aot/1512431673<strong>Homaion Roohian</strong>, <strong>Soroosh Mohammadi Farsani</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 228--236.</p><p><strong>Abstract:</strong><br/>
Let $\alpha>0$ and $\gamma>0$. We consider integral operator of the form $${\mathcal{G}}_{\phi_\gamma}f(x):=\frac{1}{\Psi_\gamma (x)}\int_0^x (1-\frac{y}{x})^{\alpha-1}\phi_\gamma(y) f(y)dy \quad x>0.$$ This paper is devoted to the study of the infinity behavior of ${\mathcal{G}}_{\phi_\gamma}$. We also provide separately result on the similar problem in the weighted Lebesgue space.
</p>projecteuclid.org/euclid.aot/1512431673_20171204185436Mon, 04 Dec 2017 18:54 ESTEquivalent conditions of a Hardy-type integral inequality related to the extended Riemann zeta functionhttps://projecteuclid.org/euclid.aot/1512431674<strong>Michael Th Rassias</strong>, <strong>Bicheng Yang</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 237--256.</p><p><strong>Abstract:</strong><br/>
By the use of techniques of real analysis and weight functions, we obtain two lemmas and build a few equivalent conditions of a Hardy-type integral inequality with a non-homogeneous kernel, related to a parameter where the constant factor is expressed in terms of the extended Riemann zeta function. Meanwhile, a few equivalent conditions for two kinds of Hardy-type integral inequalities with the homogeneous kernel are deduced. We also consider the operator expressions.
</p>projecteuclid.org/euclid.aot/1512431674_20171204185436Mon, 04 Dec 2017 18:54 ESTExistence theorems for attractive points of semigroups of Bregman generalized nonspreading mappings in Banach spaceshttps://projecteuclid.org/euclid.aot/1512431675<strong>Bashir Ali</strong>, <strong>Murtala Haruna Harbau</strong>, <strong>Lawan Haruna Yusuf</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 257--268.</p><p><strong>Abstract:</strong><br/>
In this paper, we establish new attractive point theorems for semigroups of generalized Bregman nonspreading mappings in reflexive Banach spaces. Our theorems improve and extend many results announced recently in the literature.
</p>projecteuclid.org/euclid.aot/1512431675_20171204185436Mon, 04 Dec 2017 18:54 ESTBoundedness of multilinear integral operators and their commutators on generalized Morrey spaceshttps://projecteuclid.org/euclid.aot/1512431676<strong>Panwang Wang</strong>, <strong>Zongguang Liu</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 269--286.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain some boundedness of multilinear Calderón-Zygmund Operators, multilinear fractional integral operators and their commutators on generalized Morrey Spaces.
</p>projecteuclid.org/euclid.aot/1512431676_20171204185436Mon, 04 Dec 2017 18:54 ESTSemigroup homomorphisms on matrix algebrashttps://projecteuclid.org/euclid.aot/1512431677<strong>Bernhard Burgstaller</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 287--292.</p><p><strong>Abstract:</strong><br/>
We explore the connection between ring homomorphisms and semigroup homomorphisms on matrix algebras over rings or $C^*$-algebras. Further, we give a connection between group homomorphisms on the general linear groups of a matrix stable $C^*$-algebra and their potentially extended homomorphisms on the whole $C^*$-algebra.
</p>projecteuclid.org/euclid.aot/1512431677_20171204185436Mon, 04 Dec 2017 18:54 ESTApplications of ternary rings to $C^*$-algebrashttps://projecteuclid.org/euclid.aot/1512431678<strong>Fernando Abadie</strong>, <strong>Damián Ferraro</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 293--317.</p><p><strong>Abstract:</strong><br/>
We show that there is a functor from the category of positive admissible ternary rings to the category of $*$-algebras, which induces an isomorphism of partially ordered sets between the families of $C^*$-norms on the ternary ring and its corresponding $*$-algebra. We apply this functor to obtain Morita-Rieffel equivalence results between cross-sectional $C^*$-algebras of Fell bundles, and to extend the theory of tensor products of $C^*$-algebras to the larger category of full Hilbert $C^*$-modules. We prove that, like in the case of $C^*$-algebras, there exist maximal and minimal tensor products. As applications we give simple proofs of the invariance of nuclearity and exactness under Morita-Rieffel equivalence of $C^*$-algebras.
</p>projecteuclid.org/euclid.aot/1512431678_20171204185436Mon, 04 Dec 2017 18:54 EST$k$th-order slant Toeplitz operators on the Fock spacehttps://projecteuclid.org/euclid.aot/1512431679<strong>Shivam Kumar Kumar Singh</strong>, <strong>Anuradha Gupta</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 318--333.</p><p><strong>Abstract:</strong><br/>
The notion of slant Toeplitz operators $B_\phi$ and $k$th-order slant Toeplitz operators $B_\phi^k$ on the Fock space is introduced and some of its properties are investigated. The Berezin transform of slant Toeplitz operator $B_\phi$ is also obtained. In addition, the commutativity of $k$th-order slant Toeplitz operators with co-analytic and harmonic symbols is discussed.
</p>projecteuclid.org/euclid.aot/1512431679_20171204185436Mon, 04 Dec 2017 18:54 ESTComparison results for proper multisplittings of rectangular matriceshttps://projecteuclid.org/euclid.aot/1512431680<strong>Chinmay Kumar Giri</strong>, <strong>Debasisha Mishra</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 334--352.</p><p><strong>Abstract:</strong><br/>
The least square solution of minimum norm of a rectangular linear system of equations can be found out iteratively by using matrix splittings. However, the convergence of such an iteration scheme arising out of a matrix splitting is practically very slow in many cases. Thus, works on improving the speed of the iteration scheme have attracted great interest. In this direction, comparison of the rate of convergence of the iteration schemes produced by two matrix splittings is very useful. But, in the case of matrices having many matrix splittings, this process is time-consuming. The main goal of the current article is to provide a solution to the above issue by using proper multisplittings. To this end, we propose a few comparison theorems for proper weak regular splittings and proper nonnegative splittings first. We then derive convergence and comparison theorems for proper multisplittings with the help of the theory of proper weak regular splittings.
</p>projecteuclid.org/euclid.aot/1512431680_20171204185436Mon, 04 Dec 2017 18:54 ESTAlmost periodicity of abstract Volterra integro-differential equationshttps://projecteuclid.org/euclid.aot/1512431681<strong>Marko Kostić</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 353--382.</p><p><strong>Abstract:</strong><br/>
The main purpose of this paper is to investigate almost periodic properties of various classes of $(a,k)$-regularized $C$-resolvent families in Banach spaces. We contemplate the work of many other authors working in this field, giving also some original contributions and applications. In general case, $(a,k)$-regularized $C$-resolvent families under our considerations are degenerate and their subgenerators are multivalued linear operators or pairs of closed linear operators. We also consider the class of $(a,k)$-regularized $(C_{1},C_{2})$-existence and uniqueness families, where the operators $C_{1}$ and $C_{2}$ are not necessarily injective, and provide several illustrative examples of abstract Volterra integro-differential equations which do have almost periodic solutions.
</p>projecteuclid.org/euclid.aot/1512431681_20171204185436Mon, 04 Dec 2017 18:54 ESTA note on O-frames for operatorshttps://projecteuclid.org/euclid.aot/1512431682<strong>Chander Shekhar</strong>, <strong>Shiv Kumar Kaushik</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 3, 383--395.</p><p><strong>Abstract:</strong><br/>
A sufficient condition for a boundedly complete O-frame and a necessary condition for an unconditional O-frame are given. Also, a necessary and sufficient condition for an absolute O-frame is obtained. Finally, it is proved that if two operators have an absolute O-frame, then their product also has an absolute O-frame.
</p>projecteuclid.org/euclid.aot/1512431682_20171204185436Mon, 04 Dec 2017 18:54 ESTHomomorphic conditional expectations as noncommutative retractionshttps://projecteuclid.org/euclid.aot/1512431716<strong>Robert Pluta</strong>, <strong>Bernard Russo</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 396--408.</p><p><strong>Abstract:</strong><br/>
Let $A$ be a $C^*$-algebra and $\mathcal{E}: A \to A$ a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, $$\mathcal{E}(x)^* \mathcal{E}(x) \leq \mathcal{E}(x^* x),$$ implies that $$\Vert \mathcal{E}(x)\Vert ^2 \leq \Vert \mathcal{E}(x^* x)\Vert.$$ In this note we show that $\mathcal{E}$ is homomorphic (in the sense that $\mathcal{E}(xy) = \mathcal{E}(x)\mathcal{E}(y)$ for every $x, y$ in $A$) if and only if $$\Vert \mathcal{E}(x)\Vert^2 = \Vert \mathcal{E}(x^*x)\Vert,$$ for every $x$ in $A$. We also prove that a homomorphic conditional expectation on a commutative $C^*$-algebra $C_0(X)$ is given by composition with a continuous retraction of $X$. One may therefore consider homomorphic conditional expectations as noncommutative retractions.
</p>projecteuclid.org/euclid.aot/1512431716_20171204185525Mon, 04 Dec 2017 18:55 ESTVariants of Weyl's theorem for direct sums of closed linear operatorshttps://projecteuclid.org/euclid.aot/1512431717<strong>Anuradha Gupta</strong>, <strong>Karuna Mamtani</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 409--418.</p><p><strong>Abstract:</strong><br/>
If $T$ is an operator with compact resolvent and $S$ is any densely defined closed linear operator, then the orthogonal direct sum of $T$ and $S$ satisfies various Weyl type theorems if some necessary conditions are imposed on the operator $S$. It is shown that if $S$ is isoloid and satisfies Weyl's theorem, then $T \oplus S$ satisfies Weyl's theorem. Analogous result is proved for a-Weyl's theorem. Further, it is shown that Browder's theorem is directly transmitted from $S$ to $T \oplus S$. The converse of these results have also been studied.
</p>projecteuclid.org/euclid.aot/1512431717_20171204185525Mon, 04 Dec 2017 18:55 ESTOn orthogonal decomposition of a Sobolev spacehttps://projecteuclid.org/euclid.aot/1512431718<strong>Dejenie Lakew</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 419--427.</p><p><strong>Abstract:</strong><br/>
The theme of this short article is to investigate an orthogonal decomposition of the Sobolev space $W^{1,2}\left( \Omega \right) $ as $$ W^{1,2}\left( \Omega \right) =A^{2,2}\left( \Omega \right) \oplus D^{2}\left( W_{0}^{3,2}\left( \Omega \right) \right)$$ and look at some of the properties of the inner product therein and the distance defined from the inner product. We also determine the dimension of the orthogonal difference space $W^{1,2}\left( \Omega \right) \ominus \left(W_{0}^{1,2}\left( \Omega \right) \right) ^{\perp }$ and show the expansion of Sobolev spaces as their regularity increases.
</p>projecteuclid.org/euclid.aot/1512431718_20171204185525Mon, 04 Dec 2017 18:55 ESTOn symmetry of Birkhoff-James orthogonality of linear operatorshttps://projecteuclid.org/euclid.aot/1512431719<strong>Puja Ghosh</strong>, <strong>Debmalya Sain</strong>, <strong>Kallol Paul</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 428--434.</p><p><strong>Abstract:</strong><br/>
A bounded linear operator $T$ on a normed linear space $\mathbb{X}$ is said to be right symmetric (left symmetric) if $A \perp_{B} T \Rightarrow T \perp_B A $ ($T \perp_{B} A \Rightarrow A \perp_B T $) for all $ A \in B(\mathbb{X}),$ the space of all bounded linear operators on $\mathbb{X}$. Turnšek [Linear Algebra Appl., 407 (2005), 189-195] proved that if $\mathbb{X}$ is a Hilbert space then $T$ is right symmetric if and only if $T$ is a scalar multiple of an isometry or coisometry. This result fails in general if the Hilbert space is replaced by a Banach space. The characterization of right and left symmetric operators on a Banach space is still open. In this paper we study the orthogonality in the sense of Birkhoff-James of bounded linear operators on $(\mathbb{R}^n, \Vert \cdot \Vert_{\infty}) $ and characterize the right symmetric and left symmetric operators on $(\mathbb{R}^n, \Vert \cdot \Vert_{\infty})$.
</p>projecteuclid.org/euclid.aot/1512431719_20171204185525Mon, 04 Dec 2017 18:55 ESTTraces for fractional Sobolev spaces with variable exponentshttps://projecteuclid.org/euclid.aot/1512431720<strong>Leandro Del Pezzo</strong>, <strong>Julio D. Rossi</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 435--446.</p><p><strong>Abstract:</strong><br/>
In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if $p \colon \overline{\Omega}\times \overline{\Omega} \rightarrow (1,\infty)$ and $q \colon \partial \Omega \rightarrow (1,\infty )$ are continuous functions such that $$\frac{(n-1)p(x,x)}{n-sp(x,x)}>q(x) \qquad \mathrm{in} \hspace{1em} \partial \Omega \cap \lbrace x \in \overline{\Omega}\colon n-sp(x,x) >0 \rbrace,$$ then the inequality $$||f||_{L^{q(\cdot)}(\partial\Omega)} \leq C \left\lbrace ||f||_{L^{\bar{p}(\cdot)}(\Omega)} + [f]_{s,p ( \cdot , \cdot )} \right\rbrace $$ holds. Here $\bar{p}(x)=p(x,x)$ and $\lbrack f \rbrack_{s,p(\cdot,\cdot)} $ denotes the fractional seminorm with variable exponent, that is given by $$[f]_{s,p(\cdot , \cdot)} := \mathrm{inf} \left\lbrace \lambda > 0: \int_{\Omega}\int_{\Omega }\frac{|f(x)-f(y)|^{p(x,y)}}{\lambda ^{p(x,y)} |x-y|^{n+sp(x,y)}}dxdy \lt 1 \right\rbrace$$ and $||f||_{L^{q(\cdot)}(\partial\Omega)}$ and $||f||_{L^{\bar{p}(\cdot)}(\Omega)}$ are the usual Lebesgue norms with variable exponent.
</p>projecteuclid.org/euclid.aot/1512431720_20171204185525Mon, 04 Dec 2017 18:55 ESTStructures on the way from classical to quantum spaces and their tensor productshttps://projecteuclid.org/euclid.aot/1512431721<strong>Alexander Helemskii</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 447--467.</p><p><strong>Abstract:</strong><br/>
We study tensor products of two structures situated, in a sense, between normed spaces and (abstract) operator spaces. We call them Lambert and proto-Lambert spaces and pay more attention to the latter ones. The considered two tensor products lead to essentially different norms in the respective spaces. Moreover, the proto-Lambert tensor product is especially nice for spaces with the maximal proto-Lambert norm and in particular, for $L_1$-spaces. At the same time the Lambert tensor product is nice for Hilbert spaces with the minimal Lambert norm.
</p>projecteuclid.org/euclid.aot/1512431721_20171204185525Mon, 04 Dec 2017 18:55 ESTOn skew $[m,C]$-symmetric operatorshttps://projecteuclid.org/euclid.aot/1512431722<strong>Muneo Chō</strong>, <strong>Biljana Načevska-Nastovska</strong>, <strong>Jun Tomiyama</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 468--474.</p><p><strong>Abstract:</strong><br/>
In this paper, first we characterize the spectra of skew $[m,C]$-symmetric operators and we also prove that if operators $T$ and $S$ are $C$-doubly commuting operators, $T$ is a skew $[m,C]$-symmetric operator and $Q$ is an $n$-nilpotent operator, then $T+Q$ is a skew $[m+2n-2,C]$-symmetric operator. Finally, we show that if $T$ is skew $[m,C]$-symmetric and $S$ is $[n,D]$-symmetric, then $T \otimes S$ is skew $[m+n-1, C \otimes D]$-symmetric.
</p>projecteuclid.org/euclid.aot/1512431722_20171204185525Mon, 04 Dec 2017 18:55 ESTPseudospectra of elements of reduced Banach algebrashttps://projecteuclid.org/euclid.aot/1512431723<strong>Arundhathi Krishnan</strong>, <strong>S. H. Kulkarni</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 475--493.</p><p><strong>Abstract:</strong><br/>
Let $A$ be a Banach algebra with identity $1$ and $p \in A$ be a non-trivial idempotent. Then $q=1-p$ is also an idempotent. The subalgebras $pAp$ and $qAq$ are Banach algebras, called reduced Banach algebras, with identities $p$ and $q$ respectively. For $a \in A$ and $\varepsilon > 0$, we examine the relationship between the $\varepsilon$-pseudospectrum $\Lambda_{\varepsilon}(A,a)$ of $a \in A$, and $\varepsilon$-pseudospectra of $pap \in pAp$ and $qaq \in qAq$. We also extend this study by considering a finite number of idempotents $p_{1},\cdots,p_{n}$, as well as an arbitrary family of idempotents satisfying certain conditions.
</p>projecteuclid.org/euclid.aot/1512431723_20171204185525Mon, 04 Dec 2017 18:55 EST2-Local derivations on matrix algebras and algebras of measurable operatorshttps://projecteuclid.org/euclid.aot/1512431724<strong>Shavkat Ayupov</strong>, <strong>Karimbergen Kudaybergenov</strong>, <strong>Amir Alauadinov</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 494--505.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{A}$ be a unital Banach algebra such that any Jordan derivation from $\mathcal{A}$ into any $\mathcal{A}$-bimodule $\mathcal{M}$ is a derivation. We prove that any 2-local derivation from the algebra $M_n(\mathcal{A})$ into $M_n(\mathcal{M})$ $(n \geq 3)$ is a derivation. We apply this result to show that any 2-local derivation on the algebra of locally measurable operators affiliated with a von Neumann algebra without direct abelian summands is a derivation.
</p>projecteuclid.org/euclid.aot/1512431724_20171204185525Mon, 04 Dec 2017 18:55 ESTA formulation of the Jacobi coefficients $c^l_j(\alpha, \beta)$ via Bell polynomialshttps://projecteuclid.org/euclid.aot/1512431725<strong>Stuart Day</strong>, <strong>Ali Taheri</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 506--515.</p><p><strong>Abstract:</strong><br/>
The Jacobi polynomials $(\mathscr{P}^{(\alpha, \beta)}_k: k \ge 0, \alpha, \beta >-1)$ are deeply intertwined with the Laplacian on compact rank one symmetric spaces. They represent the spherical or zonal functions and as such constitute the main ingredients in describing the spectral measures and spectral projections associated with the Laplacian on these spaces. In this note we strengthen this connection by showing that a set of spectral and geometric quantities associated with Jacobi operator fully describe the Maclaurin coefficients associated with the heat and other related Schwartzian kernels and present an explicit formulation of these quantities using the Bell polynomials.
</p>projecteuclid.org/euclid.aot/1512431725_20171204185525Mon, 04 Dec 2017 18:55 ESTBesov-Dunkl spaces connected with generalized Taylor formula on the real linehttps://projecteuclid.org/euclid.aot/1512431726<strong>Chokri Abdelkefi</strong>, <strong>Faten Rached</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 516--530.</p><p><strong>Abstract:</strong><br/>
In the present paper, we define for the Dunkl tranlation operators on the real line, the Besov-Dunkl space of functions for which the remainder in the generalized Taylor's formula has a given order. We provide characterization of these spaces by the Dunkl convolution.
</p>projecteuclid.org/euclid.aot/1512431726_20171204185525Mon, 04 Dec 2017 18:55 ESTStability of the cosine-sine functional equation with involutionhttps://projecteuclid.org/euclid.aot/1512431727<strong>Jeongwook Chang</strong>, <strong>Chang-Kwon Choi</strong>, <strong>Jongjin Kim</strong>, <strong>Prasanna K. Sahoo</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 531--546.</p><p><strong>Abstract:</strong><br/>
Let $S$ and $G$ be a commutative semigroup and a commutative group respectively, $\Bbb C$ and $\Bbb R^+$ the sets of complex numbers and nonnegative real numbers respectively, $\sigma : S \to S$ or $\sigma : G \to G$ an involution and $\psi : G \to \Bbb R^+$ be fixed. In this paper, we first investigate general solutions of the equation $$g(x+ \sigma y)=g(x)g(y)+f(x)f(y)$$ for all $ x,y \in S$, where $f, g : S \to \Bbb C$ are unknown functions to be determined. Secondly, we consider the Hyers-Ulam stability of the equation, i.e., we study the functional inequality $$|g(x+\sigma y)-g(x)g(y)-f(x)f(y)|\le \psi(y)$$ for all $x,y \in G$, where $f, g : G \to \Bbb C$.
</p>projecteuclid.org/euclid.aot/1512431727_20171204185525Mon, 04 Dec 2017 18:55 EST$L^p$ Fourier transformation on non-unimodular locally compact groupshttps://projecteuclid.org/euclid.aot/1512431728<strong>Marianne Terp</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 2, Number 4, 547--583.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a locally compact group with modular function $\Delta$ and left regular representation $\lambda$. We define the $L^p$ Fourier transform of a function $f \in L^p(G)$, $1 \le p \le 2$, to be essentially the operator $\lambda(f)\Delta^{\frac{1}{q}}$ on $L^2(G)$ (where $\frac{1}{p}+\frac{1}{q}=1$) and show that a generalized Hausdorff-Young theorem holds. To do this, we first treat in detail the spatial $L^p$ spaces $L^p(\psi_0)$, $1 \le p \le \infty$, associated with the von Neumann algebra $M=\lambda(G)^{\prime\prime}$ on $L^2(G)$ and the canonical weight $\psi_0$ on its commutant. In particular, we discuss isometric isomorphisms of $L^2(\psi_0)$ onto $L^2(G)$ and of $L^1(\psi_0)$ onto the Fourier algebra $A(G)$. Also, we give a characterization of positive definite functions belonging to $A(G)$ among all continuous positive definite functions.
</p>projecteuclid.org/euclid.aot/1512431728_20171204185525Mon, 04 Dec 2017 18:55 ESTComplex interpolation and non-commutative integrationhttps://projecteuclid.org/euclid.aot/1512497949<strong>Klaus Werner</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 1--16.</p><p><strong>Abstract:</strong><br/>
We show that under suitable conditions interpolation between a Banach space and its dual yields a Hilbert space at $\theta =\frac{1}{2}$. By application of this result to the special case of the non-commutative $L^p$-spaces of Leinert [Int. J. Math. 2 (1991), no. 2, 177-182] and Terp [J. Operator Theory 8 (1982), 327-360] we conclude that $L^2$ is a Hilbert space and that $L^p$ is isometrically isomorphic to the dual of $L^q$ without using the isomorphisms of these spaces to $L^p$-spaces of Hilsum [J. Funct. Anal. 40 (1981), 151-169.] and Haagerup [Colloq. Internat. CNRS, 274, CNRS, Paris, 1979].\Haagerup and Pisier [Canad. J. Math. 41 (1989), no. 5, 882-906.], Pisier [Mem. Amer. Math. Soc. 122 (1996), no. 585, viii+103 pp] and Watbled [C. R. Acad. Sci. Paris, t. 321, Série I, p. 1437-1440, 1995] gave conditions under which interpolation between a Banach space and its conjugate dual yields a Hilbert space at $\frac{1}{2}$. The result mentioned above when put in “conjugate form” extends their results.
</p>projecteuclid.org/euclid.aot/1512497949_20171205131916Tue, 05 Dec 2017 13:19 ESTSemicontinuity and closed faces of $C^*$-algebrashttps://projecteuclid.org/euclid.aot/1512497950<strong>Lawrence G. Brown</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 17--41.</p><p><strong>Abstract:</strong><br/>
C. Akemann and G.K. Pedersen [Duke Math. J. 40 (1973), 785-795.] defined three concepts of semicontinuity for self-adjoint elements of $A^{**}$, the enveloping von Neumann algebra of a $C^*$-algebra $A$. We give the basic properties of the analogous concepts for elements of $pA^{**}p$, where $p$ is a closed projection in $A^{**}$. In other words, in place of affine functionals on $Q$, the quasi-state space of $A$, we consider functionals on $F(p)$, the closed face of $Q$ suppported by $p$. We prove an interpolation theorem: If $h \geq k$, where $h$ is lower semicontinuous on $F(p)$ and $k$ upper semicontinuous, then there is a continuous affine functional $x$ on $F(p)$ such that $k \leq x \leq h$. We also prove an interpolation-extension theorem: Now $h$ and $k$ are given on $Q$, $x$ is given on $F(p)$ between $h_{|F(p)}$ and $k_{|F(p)}$, and we seek to extend $x$ to $\widetilde x$ on $Q$ so that $k \leq \widetilde x \leq h$. We give a characterization of $pM(A)_{{\mathrm{sa}}}p$ in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity.
</p>projecteuclid.org/euclid.aot/1512497950_20171205131916Tue, 05 Dec 2017 13:19 ESTThe closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebrahttps://projecteuclid.org/euclid.aot/1512497951<strong>Marcel de Jeu</strong>, <strong>Jun Tomiyama</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 42--52.</p><p><strong>Abstract:</strong><br/>
If $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then an involutive Banach algebra $\ell^1(\Sigma)$ of crossed product type is naturally associated with the topological dynamical system $\Sigma=(X,\sigma)$. We initiate the study of the relation between two-sided ideals of $\ell^1(\Sigma)$ and ${\mathrm C}^*(\Sigma)$, the enveloping $\mathrm{C}^*$-algebra ${\mathrm C}(X)\rtimes_\sigma \mathbb Z$ of $\ell^1(\Sigma)$. Among others, we prove that the closure of a proper two-sided ideal of $\ell^1(\Sigma)$ in ${\mathrm C}^*(\Sigma)$ is again a proper two-sided ideal of ${\mathrm C}^*(\Sigma)$.
</p>projecteuclid.org/euclid.aot/1512497951_20171205131916Tue, 05 Dec 2017 13:19 ESTPositive map as difference of two completely positive or super-positive mapshttps://projecteuclid.org/euclid.aot/1512497952<strong>Tsuyoshi Ando</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 53--60.</p><p><strong>Abstract:</strong><br/>
For a linear map from ${\mathbb M}_m$ to ${\mathbb M}_n$, besides the usual positivity, there are two stronger notions, complete positivity and super positivity. Given a positive linear map $\varphi$ we study a decomposition $\varphi = \varphi^{(1)} - \varphi^{(2)}$ with completely positive linear maps $\varphi^{(j)} (j = 1,2)$. Here $\varphi^{(1)} + \varphi^{(2)}$ is of simple form with norm small as possible. The same problem is discussed with super-positivity in place of complete positivity.
</p>projecteuclid.org/euclid.aot/1512497952_20171205131916Tue, 05 Dec 2017 13:19 ESTSome natural subspaces and quotient spaces of $L^1$https://projecteuclid.org/euclid.aot/1512497953<strong>Gilles Godefroy</strong>, <strong>Nicolas Lerner</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 61--74.</p><p><strong>Abstract:</strong><br/>
We show that the space $\mathrm{Lip}_0(\mathbb R^n)$ is the dual space of $L^{1}({\mathbb R}^{n}; {\mathbb R}^{n})/N$ where $N$ is the subspace of $L^{1}({\mathbb R}^{n}; {\mathbb R}^{n})$ consisting of vector fields whose divergence vanishes identically. We prove that although the quotient space $L^{1}({\mathbb R}^{n}; {\mathbb R}^{n})/N$ is weakly sequentially complete, the subspace $N$ is not nicely placed - in other words, its unit ball is not closed for the topology $\tau_m$ of local convergence in measure. We prove that if $\Omega$ is a bounded open star-shaped subset of $\mathbb {R}^n$ and $X$ is a dilation-stable closed subspace of $L^1(\Omega)$ consisting of continuous functions, then the unit ball of $X$ is compact for the compact-open topology on $\Omega$. It follows in particular that such spaces $X$, when they have Grothendieck's approximation property, have unconditional finite-dimensional decompositions and are isomorphic to weak*-closed subspaces of $l^1$. Numerous examples are provided where such results apply.
</p>projecteuclid.org/euclid.aot/1512497953_20171205131916Tue, 05 Dec 2017 13:19 ESTPartial isometries: a surveyhttps://projecteuclid.org/euclid.aot/1512497954<strong>Francisco J Fernández-Polo</strong>, <strong>Antonio Peralta</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 75--116.</p><p><strong>Abstract:</strong><br/>
We survey the main results characterizing partial isometries in C$^*$-algebras and tripotents in JB$^*$-triples obtained in terms of regularity, conorm, quadratic-conorm, and the geometric structure of the underlying Banach spaces.
</p>projecteuclid.org/euclid.aot/1512497954_20171205131916Tue, 05 Dec 2017 13:19 ESTOperators with compatible ranges in an algebra generated by two orthogonal projectionshttps://projecteuclid.org/euclid.aot/1512497955<strong>Ilya M Spitkovsky</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 117--122.</p><p><strong>Abstract:</strong><br/>
The criterion is obtained for operators $A$ from the algebra generated by two orthogonal projections $P,Q$ to have a compatible range, i.e., coincide with the hermitian conjugate of $A$ on the orthogonal complement to the sum of their kernels. In the particular case of $A$ being a polynomial in $P,Q$, some easily verifiable conditions are derived.
</p>projecteuclid.org/euclid.aot/1512497955_20171205131916Tue, 05 Dec 2017 13:19 ESTPermanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin propertyhttps://projecteuclid.org/euclid.aot/1512497956<strong>Hiroyuki Osaka</strong>, <strong>Tamotsu Teruya</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 123--136.</p><p><strong>Abstract:</strong><br/>
Let $P \subset A$ be an inclusion of unital $C^*$-algebras and $E: A \rightarrow P$ be a faithful conditional expectation of index finite type. Suppose that $E$ has the Rokhlin property. Then $\mathrm{dr}(P) \leq \mathrm{dr}(A)$ and $dim_{nuc}(P) \leq dim_{nuc}(A)$. This can be applied to Rokhlin actions of finite groups. We also show that under the same above assumption if $A$ is exact and pure, that is, the Cuntz semigroups $W(A)$ has strict comparison and is almost divisible, then $P$ and the basic contruction $C^*\langle A, e_P \rangle$ are also pure.
</p>projecteuclid.org/euclid.aot/1512497956_20171205131916Tue, 05 Dec 2017 13:19 ESTAlmost Hadamard matrices with complex entrieshttps://projecteuclid.org/euclid.aot/1512497957<strong>Teodor Banica</strong>, <strong>Ion Nechita</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 137--177.</p><p><strong>Abstract:</strong><br/>
We discuss an extension of the almost Hadamard matrix formalism, to the case of complex matrices. Quite surprisingly, the situation here is very different from the one in the real case, and our conjectural conclusion is that there should be no such matrices, besides the usual Hadamard ones. We verify this conjecture in a number of situations, and notably for most of the known examples of real almost Hadamard matrices, and for some of their complex extensions. We discuss as well some potential applications of our conjecture, to the general study of complex Hadamard matrices.
</p>projecteuclid.org/euclid.aot/1512497957_20171205131916Tue, 05 Dec 2017 13:19 ESTNon-commutative rational functions in strong convergent random variableshttps://projecteuclid.org/euclid.aot/1512497958<strong>Sheng Yin</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 178--192.</p><p><strong>Abstract:</strong><br/>
Random matrices like GUE, GOE and GSE have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and Thorbørnsen in their paper in 2005, it is called strong convergence property and then more random matrices with this property are followed. In general, the definition can be stated for a sequence of tuples over some $\mathrm{C}^*$-algebras. In this paper, we want to show that, for a sequence of strongly convergent random variables, non-commutative polynomials can be extended to non-commutative rational functions under certain assumptions. As a direct corollary, we can conclude that for a tuple $(X_{1}^{\left(n\right)},\cdots,X_{m}^{\left(n\right)})$ of independent GUE random matrices, $r(X_{1}^{\left(n\right)},\cdots,X_{m}^{\left(n\right)})$ converges in trace and in norm to $r(s_{1},\cdots,s_{m})$ almost surely, where $r$ is a rational function and $(s_{1},\cdots,s_{m})$ is a tuple of freely independent semi-circular elements which lies in the domain of $r$.
</p>projecteuclid.org/euclid.aot/1512497958_20171205131916Tue, 05 Dec 2017 13:19 ESTFourier multiplier norms of spherical functions on the generalized Lorentz groupshttps://projecteuclid.org/euclid.aot/1512497959<strong>Troels Steenstrup</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 193--230.</p><p><strong>Abstract:</strong><br/>
Our main result provides a closed expression for the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups $SO_0(1,n)$ (for $n \geq 2$). As a corollary, we find that there is no uniform bound on the completely bounded Fourier multiplier norm of the spherical functions on the generalized Lorentz groups. We extend the latter result to the groups $SU(1,n)$, $Sp(1,n)$ (for $n \geq 2$) and the exceptional group $F_{4(-20)}$, and as an application we obtain that each of the above mentioned groups has a completely bounded Fourier multiplier, which is not the coefficient of a uniformly bounded representation of the group on a Hilbert space.
</p>projecteuclid.org/euclid.aot/1512497959_20171205131916Tue, 05 Dec 2017 13:19 ESTOn a class of Banach algebras associated to harmonic analysis on locally compact groups and semigroupshttps://projecteuclid.org/euclid.aot/1512497960<strong>Anthony To-Ming Lau</strong>, <strong>Hung Le Pham</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 231--246.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to present some old and recent results for the class of $F$-algebras which include most classes of Banach algebras that are important in abstract harmonic analysis. We also introduce a subclass of the class of $F$-algebras, called normal $F$-algebras, that captures better the measure algebras and the (reduced) Fourier-Stieltjes algebras, and use this to give new characterisations the reduced Fourier-Stieltjes algebras of discrete groups.
</p>projecteuclid.org/euclid.aot/1512497960_20171205131916Tue, 05 Dec 2017 13:19 ESTUniformly bounded representations and completely bounded multipliers of $\mathrm {SL}(2,\mathbb{R})$https://projecteuclid.org/euclid.aot/1512497961<strong>Francesca Astengo</strong>, <strong>Michael G. Cowling</strong>, <strong>Bianca Di Blasio</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 247--270.</p><p><strong>Abstract:</strong><br/>
We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of $\mathrm {SL}(2,\mathbb{R})$ as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal.
</p>projecteuclid.org/euclid.aot/1512497961_20171205131916Tue, 05 Dec 2017 13:19 ESTCompletely positive contractive maps and partial isometrieshttps://projecteuclid.org/euclid.aot/1512497962<strong>Berndt Brenken</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 271--294.</p><p><strong>Abstract:</strong><br/>
Associated with a completely positive contractive map $\varphi$ of a $C^*$-algebra $A$ is a universal $C^*$-algebra generated by the $C^*$-algebra $A$ along with a contraction implementing $\varphi$. We prove a dilation theorem: the map $\varphi$ may be extended to a completely positive contractive map of an augmentation of $A$. The associated $C^*$-algebra of the augmented system contains the original universal $C^*$-algebra as a corner, and the extended completely positive contractive map is implemented by a partial isometry.
</p>projecteuclid.org/euclid.aot/1512497962_20171205131916Tue, 05 Dec 2017 13:19 ESTUffe Haagerup - his life and mathematicshttps://projecteuclid.org/euclid.aot/1512497963<strong>Mohammad Sal Moslehian</strong>, <strong>Erling Størmer</strong>, <strong>Steen Thorbjørnsen</strong>, <strong>Carl Winsløw</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 1, 295--325.</p><p><strong>Abstract:</strong><br/>
In remembrance of Professor Uffe Valentin Haagerup (1949-2015), as a brilliant mathematician, we review some aspects of his life, and his outstanding mathematical accomplishments.
</p>projecteuclid.org/euclid.aot/1512497963_20171205131916Tue, 05 Dec 2017 13:19 ESTOn different type of fixed point theorem for multivalued mappings via measure of noncompactnesshttps://projecteuclid.org/euclid.aot/1513328633<strong>Nour El Houda Bouzara</strong>, <strong>Vatan Karakaya</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 2, 1--11.</p><p><strong>Abstract:</strong><br/>
In this paper by using the measure of noncompactness concept, we present new fixed point theorems for multivalued maps. In further we introduce a new class of mappings which are general than Meir-Keeler mappings. Finally, we use these results to investigate the existence of weak solutions to an Evolution differential inclusion with lack of compactness.
</p>projecteuclid.org/euclid.aot/1513328633_20171215040405Fri, 15 Dec 2017 04:04 ESTSingular Riesz measures on symmetric coneshttps://projecteuclid.org/euclid.aot/1513328634<strong>Abdelhamid Hassairi</strong>, <strong>Sallouha Lajmi</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 2, 12--25.</p><p><strong>Abstract:</strong><br/>
A fondamental theorem due to Gindikin [Russian Math. Surveys, 29 (1964), 1-89] says that the generalized power $\Delta_{s}(-\theta^{-1})$ defined on a symmetric cone is the Laplace transform of a positive measure $R_{s}$ if and only if $s$ is in a given subset $\Xi$ of $\mathbb{R}^{r}$, where $r$ is the rank of the cone. When $s$ is in a well defined part of $\Xi$, the measure $R_{s}$ is absolutely continuous with respect to Lebesgue measure and has a known expression. For the other elements $s$ of $\Xi$, the measure $R_{s}$ is concentrated on the boundary of the cone and it has never been explicitly determined. The aim of the present paper is to give an explicit description of the measure $R_{s}$ for all $s$ in $\Xi$. The work is motivated by the importance of these measures in probability theory and in statistics since they represent a generalization of the class of measures generating the famous Wishart probability distributions.
</p>projecteuclid.org/euclid.aot/1513328634_20171215040405Fri, 15 Dec 2017 04:04 ESTCover topologies, subspaces, and quotients for some spaces of vector-valued functionshttps://projecteuclid.org/euclid.aot/1513328635<strong>Terje Hõim</strong>, <strong>D. A. Robbins</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 2, 26--39.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a completely regular Hausdorff space, and let $\mathcal{D}$ be a cover of $X$ by $C_{b}$-embedded sets. Let $\pi: \mathcal{E} \rightarrow X$ be a bundle of Banach spaces (algebras), and let $\Gamma(\pi)$ be the section space of the bundle $\pi.$ Denote by $\Gamma _{b}(\pi,\mathcal{D})$ the subspace of $\Gamma (\pi )$ consisting of sections which are bounded on each $D \in \mathcal{D}$. We construct a bundle $\rho ^{\prime }: \mathcal{F}^{\prime}\rightarrow \beta X$ such that $\Gamma _{b}(\pi , \mathcal{D}) $ is topologically and algebraically isomorphic to $\Gamma(\rho^\prime)$, and use this to study the subspaces (ideals) and quotients resulting from endowing $\Gamma _{b}(\pi,\mathcal{D})$ with the cover topology determined by $\mathcal{D}$.
</p>projecteuclid.org/euclid.aot/1513328635_20171215040405Fri, 15 Dec 2017 04:04 ESTIntegral representations and asymptotic behaviour of a Mittag-Leffler type function of two variableshttps://projecteuclid.org/euclid.aot/1513328636<strong>Christian Lavault</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 2, 40--48.</p><p><strong>Abstract:</strong><br/> Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in various conditions and cases. The present paper explores the integral representations of a special function extending to two variables the two-parametric Mittag-Leffler type function. Integral representations of this functions within different variation ranges of its arguments for certain values of the parameters are thus obtained. Asymptotic expansion formulas and asymptotic properties of this function are also established for large values of the variables. This yields corresponding theorems providing integral representations as well as expansion formulas. </p>projecteuclid.org/euclid.aot/1513328636_20171215040405Fri, 15 Dec 2017 04:04 ESTOperator algebras associated to modules over an integral domainhttps://projecteuclid.org/euclid.aot/1513328637<strong>Benton Duncan</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 2, 49--62.</p><p><strong>Abstract:</strong><br/>
We use the Fock semicrossed product to define an operator algebra associated to a module over an integral domain. We consider the $C^*$-envelope of the semicrossed product, and then consider properties of these algebras as models for studying general semicrossed products.
</p>projecteuclid.org/euclid.aot/1513328637_20171215040405Fri, 15 Dec 2017 04:04 ESTOn the truncated two-dimensional moment problemhttps://projecteuclid.org/euclid.aot/1513328638<strong>Sergey Zagorodnyuk</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 2, 63--74.</p><p><strong>Abstract:</strong><br/>
We study the truncated two-dimensional moment problem (with rectangular data) to find a non-negative measure $\mu(\delta)$, $\delta\in\mathfrak{B}(\mathbb{R}^2)$, such that $\int_{\mathbb{R}^2} x_1^m x_2^n d \mu = s_{m,n}$, $0\leq m\leq M,\quad 0\leq n\leq N$, where $\{ s_{m,n} \}_{0\leq m\leq M, 0\leq n\leq N}$ is a prescribed sequence of real numbers; $M,N\in\mathbb{Z}_+$. For the cases $M=N=1$ and $M=1, N=2$ explicit numerical necessary and sufficient conditions for the solvability of the moment problem are given. In the cases $M=N=2$; $M=2, N=3$; $M=3, N=2$; $M=3, N=3$ some explicit numerical sufficient conditions for the solvability are obtained. In all the cases some solutions (not necessarily atomic) of the moment problem can be constructed.
</p>projecteuclid.org/euclid.aot/1513328638_20171215040405Fri, 15 Dec 2017 04:04 ESTThe compactness of a class of radial operators on weighted Bergman spaceshttps://projecteuclid.org/euclid.aot/1513328639<strong>Yucheng Li</strong>, <strong>Maofa Wang</strong>, <strong>Wenhua Lan</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 2, 75--85.</p><p><strong>Abstract:</strong><br/>
In this paper, we study some connection between the compactness of radial operators and the boundary behavior of the corresponding Berezin transform on weighted Bergman spaces. More precisely, we prove that, under some mild condition, the vanishing of the Berezin transform on the unit circle is equivalent to the compactness of a class of radial operators on weighted Bergman spaces. Moreover, we also study the radial essential commutant of the Toeplitz operator $T_z$.
</p>projecteuclid.org/euclid.aot/1513328639_20171215040405Fri, 15 Dec 2017 04:04 ESTExtensions of theory of regular and weak regular splittings to singular matriceshttps://projecteuclid.org/euclid.aot/1513328640<strong>Litismita Jena</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 2, 86--97.</p><p><strong>Abstract:</strong><br/>
Matrix splittings are useful in finding a solution of linear systems of equations, iteratively. In this note, we present some more convergence and comparison results for recently introduced matrix splittings called index-proper regular and index-proper weak regular splittings. We then apply to theory of double index-proper splittings.
</p>projecteuclid.org/euclid.aot/1513328640_20171215040405Fri, 15 Dec 2017 04:04 ESTOn linear maps preserving certain pseudospectrum and condition spectrum subsetshttps://projecteuclid.org/euclid.aot/1513328641<strong>Sayda Ragoubi</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 2, 98--107.</p><p><strong>Abstract:</strong><br/> We define two new types of spectrum, called the $\varepsilon$-left (or right) pseudospectrum and the $\varepsilon$-left (or right) condition spectrum, of an element $a$ in a complex unital Banach algebra $A$. We prove some basic properties among them the property that the $\varepsilon$-left (or right) condition spectrum is a particular case of Ransford spectrum. We study also the linear preserver problem for our defined functions and we establish the following: (1) Let $A$ and $B$ be complex unital Banach algebras and $\varepsilon>0$. Let $\phi : A \longrightarrow B$ be an $\varepsilon$-left (or right) pseudospectrum preserving onto linear map. Then $\phi$ preserves certain standard spectral functions. (2) Let $A$ and $B$ be complex unital Banach algebras and $0 \lt \varepsilon \lt 1$. Let $\phi : A \longrightarrow B $ be a unital linear map. Then (a) If $\phi$ is an $\varepsilon$-almost multiplicative map, then $\sigma^{l}(\phi(a))\subseteq \sigma^{l}_\varepsilon(a)$ and $\sigma^{r}(\phi(a))\subseteq \sigma^{r}_\varepsilon(a)$, for all $a \in A$. (b) If $\phi$ is an $\varepsilon$-left (or right) condition spectrum preserving, then (i) if $A$ is semi-simple, then $\phi$ is injective; (ii) if B is spectrally normed, then $\phi$ is continuous. </p>projecteuclid.org/euclid.aot/1513328641_20171215040405Fri, 15 Dec 2017 04:04 ESTCertain geometric structures of $\Lambda$-sequence spaceshttps://projecteuclid.org/euclid.aot/1513328642<strong>Atanu Manna</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 2, 108--125.</p><p><strong>Abstract:</strong><br/>
The $\Lambda$-sequence spaces $\Lambda_p$ for $1 \lt p\leq\infty$ and their generalized forms $\Lambda_{\hat{p}}$ for $1 \lt \hat{p} \lt \infty$, $\hat{p}=(p_n)$, $n\in \mathbb{N}_0$ are introduced. The James constants and strong $n$-th James constants of $\Lambda_p$ for $1 \lt p \leq \infty$ are determined. It is proved that the generalized $\Lambda$-sequence space $\Lambda_{\hat{p}}$ is a closed subspace of the Nakano sequence space $l_{\hat{p}}(\mathbb{R}^{n+1})$ of finite dimensional Euclidean space $\mathbb{R}^{n+1}$, $n\in \mathbb{N}_0$. Hence it follows that sequence spaces $\Lambda_p$ and $\Lambda_{\hat{p}}$ possess the uniform Opial property, ($\beta$)-property of Rolewicz, and weak uniform normal structure. Moreover, it is established that $\Lambda_{\hat{p}}$ possesses the coordinate wise uniform Kadec–Klee property. Further, necessary and sufficient conditions for element $x\in S(\Lambda_{\hat{p}})$ to be an extreme point of $B(\Lambda_{\hat{p}})$ are derived. Finally, estimation of von Neumann-Jordan and James constants of two dimensional $\Lambda$-sequence space $\Lambda_2^{(2)}$ are carried out. Upper bound for the Hausdorff matrix operator norm on the non-absolute type $\Lambda$-sequence spaces is also obtained.
</p>projecteuclid.org/euclid.aot/1513328642_20171215040405Fri, 15 Dec 2017 04:04 EST