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 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 ESTLinear preservers of two-sided right matrix majorization on $\mathbb{R}_{n}$https://projecteuclid.org/euclid.aot/1513876632<strong>Ahmad Mohammadhasani</strong>, <strong>Asma Ilkhanizadeh Manesh</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 451--458.</p><p><strong>Abstract:</strong><br/>
A nonnegative real matrix $R \in \mathrm {M}_{n,m}$ with the property that all its row sums are one is said to be row stochastic. For $x, y \in \mathbb{R}_{n}$, we say $x$ is right matrix majorized by $y$ (denoted by $x \prec_{r} y$) if there exists an $n$-by-$n$ row stochastic matrix $R$ such that $x = yR$. The relation $\sim_{r}$ on $\mathbb{R}_{n}$ is defined as follows. $x \sim_{r}y$ if and only if $x \prec_{r} y \prec_{r} x$. In the present paper, we characterize the linear preservers of $\sim_{r}$ on $\mathbb{R}_{n}$, and answer the question raised by F. Khalooei [Wavelet Linear Algebra 1 (2014), no. 1, 43-50].
</p>projecteuclid.org/euclid.aot/1513876632_20180426220054Thu, 26 Apr 2018 22:00 EDTPompeiu-Čebyšev type inequalities for selfadjoint operators in Hilbert spaceshttps://projecteuclid.org/euclid.aot/1513876633<strong>Mohammad W. Alomari</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 459--472.</p><p><strong>Abstract:</strong><br/>
In this work, generalizations of some inequalities for continuous $h$-synchronous ($h$-asynchronous) functions of selfadjoint linear operators in Hilbert spaces are proved.
</p>projecteuclid.org/euclid.aot/1513876633_20180426220054Thu, 26 Apr 2018 22:00 EDTPerturbation of minimum attaining operatorshttps://projecteuclid.org/euclid.aot/1518016380<strong>Jadav Ganesh</strong>, <strong>Golla Ramesh</strong>, <strong>Daniel Sukumar</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 473--490.</p><p><strong>Abstract:</strong><br/>
We prove that the minimum attaining property of a bounded linear operator on a Hilbert space $H$ whose minimum modulus lies in the discrete spectrum, is stable under small compact perturbations. We also observe that given a bounded operator with strictly positive essential minimum modulus, the set of compact perturbations which fail to produce a minimum attaining operator is smaller than a nowhere dense set. In fact, it is a porous set in the ideal of all compact operators on $H$. Further, we try to extend these stability results to perturbations by all bounded linear operators with small norm and obtain subsequent results.
</p>projecteuclid.org/euclid.aot/1518016380_20180426220054Thu, 26 Apr 2018 22:00 EDTBesicovitch almost automorphic solutions of nonautonomous differential equations of first orderhttps://projecteuclid.org/euclid.aot/1518016381<strong>Marko Kostić</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 491--506.</p><p><strong>Abstract:</strong><br/>
The main purpose of this paper is to analyze the existence and uniqueness of Besicovitch almost automorphic solutions and weighted Besicovitch pseudo-almost automorphic solutions of nonautonomous differential equations of first order. We provide an interesting application of our abstract theoretical results.
</p>projecteuclid.org/euclid.aot/1518016381_20180426220054Thu, 26 Apr 2018 22:00 EDTA Kakutani–Mackey-like theoremhttps://projecteuclid.org/euclid.aot/1518016382<strong>Marina Haralampidou</strong>, <strong>Konstantinos Tzironis</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 507--521.</p><p><strong>Abstract:</strong><br/>
We give a partial extension of a Kakutani–Mackey theorem for quasi-complemented vector spaces. This can be applied in the representation theory of certain complemented (non-normed) topological algebras. The existence of continuous linear maps, in the context of quasi-complemented vector spaces, is a very important issue in their study. Relative to this, we prove that every Hausdorff quasi-complemented locally convex space has continuous linear maps, under which a certain quasi-complemented locally convex space turns to be pre-Hilbert.
</p>projecteuclid.org/euclid.aot/1518016382_20180426220054Thu, 26 Apr 2018 22:00 EDT$T1$ theorem for inhomogeneous Triebel–Lizorkin and Besov spaces on RD-spaces and its applicationhttps://projecteuclid.org/euclid.aot/1518016383<strong>Fanghui Liao</strong>, <strong>Zongguang Liu</strong>, <strong>Hongbin Wang</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 522--537.</p><p><strong>Abstract:</strong><br/>
Using Calderón's reproducing formulas and almost orthogonal estimates, the $T1$ theorem for the inhomogeneous Triebel–Lizorkin and Besov spaces on RD-spaces is obtained. As an application, new characterizations for these spaces with “half” the usual conditions of the approximate to the identity are presented.
</p>projecteuclid.org/euclid.aot/1518016383_20180426220054Thu, 26 Apr 2018 22:00 EDTFixed points of a class of unitary operatorshttps://projecteuclid.org/euclid.aot/1520046044<strong>Namita Das</strong>, <strong>Jitendra Kumar Behera</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 538--550.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider a class of unitary operators defined on the Bergman space of the right half plane and characterize the fixed points of these unitary operators. We also discuss certain intertwining properties of these operators. Applications of these results are also obtained.
</p>projecteuclid.org/euclid.aot/1520046044_20180426220054Thu, 26 Apr 2018 22:00 EDTWell-posedness issues for a class of coupled nonlinear Schrödinger equations with critical exponential growthhttps://projecteuclid.org/euclid.aot/1520046045<strong>Hanen Hezzi</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 551--581.</p><p><strong>Abstract:</strong><br/>
The initial value problem for some coupled nonlinear Schrödinger equations in two space dimensions with exponential growth is investigated. In the defocusing case, global well-posedness and scattering are obtained. In the focusing sign, global and nonglobal existence of solutions are discussed via potential well-method.
</p>projecteuclid.org/euclid.aot/1520046045_20180426220054Thu, 26 Apr 2018 22:00 EDTClosedness and invertibility for the sum of two closed operatorshttps://projecteuclid.org/euclid.aot/1520046046<strong>Nikolaos Roidos</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 582--605.</p><p><strong>Abstract:</strong><br/>
We show a Kalton–Weis type theorem for the general case of noncommuting operators. More precisely, we consider sums of two possibly noncommuting linear operators defined in a Banach space such that one of the operators admits a bounded $H^\infty$-calculus, the resolvent of the other one satisfies some weaker boundedness condition and the commutator of their resolvents has certain decay behavior with respect to the spectral parameters. Under this consideration, we show that the sum is closed and that after a sufficiently large positive shift it becomes invertible and moreover sectorial. As an application we recover a classical result on the existence, uniqueness, and maximal $L^{p}$-regularity for solutions of the abstract linear nonautonomous parabolic problem.
</p>projecteuclid.org/euclid.aot/1520046046_20180426220054Thu, 26 Apr 2018 22:00 EDTParallel iterative methods for solving the common null point problem in Banach spaceshttps://projecteuclid.org/euclid.aot/1520046047<strong>Truong Minh Tuyen</strong>, <strong>Nguyen Minh Trang</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 606--619.</p><p><strong>Abstract:</strong><br/>
We consider the common null point problem in Banach spaces. Then, using the hybrid projection method and the $\varepsilon $-enlargement of maximal monotone operators, we prove two strong convergence theorems for finding a solution of this problem.
</p>projecteuclid.org/euclid.aot/1520046047_20180426220054Thu, 26 Apr 2018 22:00 EDTComplex isosymmetric operatorshttps://projecteuclid.org/euclid.aot/1520046048<strong>Muneo Chō</strong>, <strong>Ji Eun Lee</strong>, <strong>T. Prasad</strong>, <strong>Kôtarô Tanahashi</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 620--631.</p><p><strong>Abstract:</strong><br/>
In this paper, we introduce complex isosymmetric and $(m,n,C)$-isosymmetric operators on a Hilbert space $\mathcal H$ and study properties of such operators. In particular, we prove that if $T \in {\mathcal B}(\mathcal H)$ is an $(m,n,C)$-isosymmetric operator and $N$ is a $k$-nilpotent operator such that $T$ and $N$ are $C$-doubly commuting, then $T + N$ is an $(m+2k-2, n+2k-1,C)$-isosymmetric operator. Moreover, we show that if $T$ is $(m,n,C)$-isosymmetric and if $S$ is $(m',D)$-isometric and $n'$-complex symmetric with a conjugation $D$, then $T \otimes S$ is $(m+m'-1,n+n'-1,C \otimes D)$-isosymmetric.
</p>projecteuclid.org/euclid.aot/1520046048_20180426220054Thu, 26 Apr 2018 22:00 EDTVariant versions of the Lewent type determinantal inequalityhttps://projecteuclid.org/euclid.aot/1522807280<strong>Ali Morassaei</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 632--638.</p><p><strong>Abstract:</strong><br/>
In this paper, we present a refinement of the Lewent determinantal inequality and show that the following inequality holds $$\det\frac{I_{\mathcal{H}}+A_1}{I_{\mathcal{H}}-A_1}+\det\frac{I_{\mathcal{H}}+A_n}{I_{\mathcal{H}}-A_n}-\sum_{j=1}^n\lambda_j \det\left(\frac{I_{\mathcal{H}}+A_j}{I_{\mathcal{H}}-A_j}\right)$$ $$\ge \det\left[\left(\frac{I_{\mathcal{H}}+A_1}{I_{\mathcal{H}}-A_1}\right)\left(\frac{I_{\mathcal{H}}+A_n}{I_{\mathcal{H}}-A_n}\right)\prod_{j=1}^n \left(\frac{I_{\mathcal{H}}+A_j}{I_{\mathcal{H}}-A_j}\right)^{-\lambda_j}\right],$$ where $A_j\in\mathbb{B}(\mathcal{H})$, $0\le A_j < I_\mathcal{H}$, $A_j$'s are trace class operators and $A_1 \le A_j \le A_n~(j=1,\ldots,n)$ and $\sum_{j=1}^n\lambda_j=1,~ \lambda_j \ge 0~ (j=1,\ldots,n)$. In addition, we present some new versions of the Lewent type determinantal inequality.
</p>projecteuclid.org/euclid.aot/1522807280_20180426220054Thu, 26 Apr 2018 22:00 EDTwUR modulus and normal structure in Banach spaceshttps://projecteuclid.org/euclid.aot/1522807281<strong>Ji Gao</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 639--646.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a Banach space. In this paper, we study the properties of wUR modulus of $X$, $\delta_X(\varepsilon, f),$ where $0 \le \varepsilon \le 2$ and $f \in S(X^*),$ and the relationship between the values of wUR modulus and reflexivity, uniform nonsquareness and normal structure, respectively. Among other results, we proved that if $ \delta_X(1, f)> 0$, for any $f\in S(X^*)$, then $X$ has weak normal structure.
</p>projecteuclid.org/euclid.aot/1522807281_20180426220054Thu, 26 Apr 2018 22:00 EDTThe matrix power means and interpolationshttps://projecteuclid.org/euclid.aot/1522807282<strong>Trung Hoa Dinh</strong>, <strong>Raluca Dumitru</strong>, <strong>Jose A. Franco</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 647--654.</p><p><strong>Abstract:</strong><br/>
It is well-known that the Heron mean is a linear interpolation between the arithmetic and the geometric means while the matrix power mean $P_t(A,B):= A^{1/2}\left(\frac{I+(A^{-1/2}BA^{-1/2})^t}{2}\right)^{1/t}A^{1/2}$ interpolates between the harmonic, the geometric, and the arithmetic means. In this article, we establish several comparisons between the matrix power mean, the Heron mean, and the Heinz mean. Therefore, we have a deeper understanding about the distribution of these matrix means.
</p>projecteuclid.org/euclid.aot/1522807282_20180426220054Thu, 26 Apr 2018 22:00 EDT$C^*$-algebra distance filtershttps://projecteuclid.org/euclid.aot/1522807283<strong>Tristan Bice</strong>, <strong>Alessandro Vignati</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 655--681.</p><p><strong>Abstract:</strong><br/>
We use nonsymmetric distances to give a self-contained account of $C^*$-algebra filters and their corresponding compact projections, simultaneously simplifying and extending their general theory.
</p>projecteuclid.org/euclid.aot/1522807283_20180426220054Thu, 26 Apr 2018 22:00 EDTOn Neugebauer's covering theoremhttps://projecteuclid.org/euclid.aot/1522807284<strong>Jésus M. Aldaz</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 682--689.</p><p><strong>Abstract:</strong><br/>
We present a new proof of a covering theorem of C. J. Neugebauer, stated in a slightly more general form than the original version; we also give an application to restricted weak type (1,1) inequalities for the uncentered maximal operator.
</p>projecteuclid.org/euclid.aot/1522807284_20180426220054Thu, 26 Apr 2018 22:00 EDTThe existence of hyperinvariant subspaces for weighted shift operatorshttps://projecteuclid.org/euclid.aot/1522807285<strong>Hossein Sadeghi</strong>, <strong>Farzollah Mirzapour</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 690--698.</p><p><strong>Abstract:</strong><br/>
We introduce some classes of Banach spaces for which the hyperinvariant subspace problem for the shift operator has positive answer. Moreover, we provide sufficient conditions on weights which ensure that certain subspaces of $\ell^2_{{\beta}}(\mathbb{Z})$ are closed under convolution. Finally we consider some cases of weighted spaces for which the problem remains open.
</p>projecteuclid.org/euclid.aot/1522807285_20180426220054Thu, 26 Apr 2018 22:00 EDTOrthogonality of bounded linear operators on complex Banach spaceshttps://projecteuclid.org/euclid.aot/1522807286<strong>Kallol Paul</strong>, <strong>Debmalya Sain</strong>, <strong>Arpita Mal</strong>, <strong>Kalidas Mandal</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 699--709.</p><p><strong>Abstract:</strong><br/>
We study Birkhoff-James orthogonality of compact linear operators on complex reflexive Banach spaces and obtain its characterization. By means of introducing new definitions, we illustrate that it is possible in the complex case, to develop a study of orthogonality of compact linear operators, analogous to the real case. Furthermore, earlier operator theoretic characterizations of Birkhoff-James orthogonality in the real case, can be obtained as simple corollaries to our present study. In fact, we obtain more than one equivalent characterizations of Birkhoff-James orthogonality of compact linear operators in the complex case, in order to distinguish the complex case from the real case.
</p>projecteuclid.org/euclid.aot/1522807286_20180426220054Thu, 26 Apr 2018 22:00 EDTAffine actions and the Yang–Baxter equationhttps://projecteuclid.org/euclid.aot/1524794448<strong>Dilian Yang</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 710--730.</p><p><strong>Abstract:</strong><br/>
In this paper, the relations between the Yang–Baxter equation and affine actions are explored in detail. In particular, we classify the injective set-theoretic solutions of the Yang–Baxter equation in two ways: (i) by their associated affine actions of their structure groups on their derived structure groups, and (ii) by the $C^*$-dynamical systems obtained from their associated affine actions. On the way to our main results, several other useful results are also obtained.
</p>projecteuclid.org/euclid.aot/1524794448_20180426220054Thu, 26 Apr 2018 22:00 EDTCharacterizing projections among positive operators in the unit spherehttps://projecteuclid.org/euclid.aot/1524794449<strong>Antonio M. Peralta</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 3, 731--744.</p><p><strong>Abstract:</strong><br/>
Let $E$ and $P$ be subsets of a Banach space $X$, and let us define the unit sphere around $E$ in $P$ as the set $$Sph(E;P) :=\left\{ x\in P : \|x-b\|=1 \hbox{ for all } b\in E \right\}.$$ Given a $C^*$-algebra $A$ and a subset $E\subset A,$ we shall write $Sph ^{+} (E)$ or $Sph ^{+}_{A} (E)$ for the set $Sph(E;S(A^+)),$ where $S(A^+)$ denotes the unit sphere of $A^+$. We prove that, for every complex Hilbert space $H$, the following statements are equivalent for every positive element $a$ in the unit sphere of $B(H)$: (a) $a$ is a projection; (b) $Sph^+_{B(H)} \left( Sph^+_{B(H)}(\{a\}) \right) =\{a\}$. We also prove that the equivalence remains true when $B(H)$ is replaced with an atomic von Neumann algebra or with $K(H_2)$, where $H_2$ is an infinite-dimensional and separable complex Hilbert space.
</p>projecteuclid.org/euclid.aot/1524794449_20180426220054Thu, 26 Apr 2018 22:00 EDT$L^p$-Hardy-Rellich and uncertainty principle inequalities on the spherehttps://projecteuclid.org/euclid.aot/1525917618<strong>Abimbola Abolarinwa</strong>, <strong>Timothy Apata</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 4, 745--762.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the Hardy-Rellich type inequalities and uncertainty principle on the geodesic sphere. Firstly, we derive $L^p$-Hardy inequalities via divergence theorem, which are in turn used to establish the $L^p$-Rellich inequalities. We also establish Heisenberg uncertainty principle on the sphere via the Hardy-Rellich type inequalities. The best constants appearing in the inequalities are shown to be sharp.
</p>projecteuclid.org/euclid.aot/1525917618_20180919220102Wed, 19 Sep 2018 22:01 EDTEstimations of the Lehmer mean by the Heron mean and their generalizations involving refined Heinz operator inequalitieshttps://projecteuclid.org/euclid.aot/1525917621<strong>Masatoshi Ito</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 4, 763--780.</p><p><strong>Abstract:</strong><br/>
As generalizations of the arithmetic and the geometric means, for positive real numbers $a$ and $b$, the power difference mean $J_{q}(a,b)=\frac{q}{q+1}\frac{a^{q+1}-b^{q+1}}{a^{q}-b^{q}}$, the Lehmer mean $L_{q}(a,b)=\frac{a^{q+1}+b^{q+1}}{a^{q}+b^{q}}$ and the Heron mean $K_{q}(a,b)=(1-q)\sqrt{ab}+q\frac{a+b}{2}$ are well known.
In this paper, concerning our recent results on estimations of the power difference mean, we obtain the greatest value $\alpha=\alpha(q)$ and the least value $\beta=\beta(q)$ such that the double inequality for the Lehmer mean $$K_{\alpha}(a,b)< L_{q}(a,b)< K_{\beta}(a,b)$$ holds for any $q \in \mathbb{R}$. We also obtain an operator version of this estimation. Moreover, we discuss generalizations of the results on estimations of the power difference and the Lehmer means.This argument involves refined Heinz operator inequalities by Liang and Shi.
</p>projecteuclid.org/euclid.aot/1525917621_20180919220102Wed, 19 Sep 2018 22:01 EDTOn behavior of Fourier coefficients and uniform convergence of Fourier series in the Haar systemhttps://projecteuclid.org/euclid.aot/1528444822<strong>M. G. Grigoryan</strong>, <strong>A. Kh. Kobelyan</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 4, 781--793.</p><p><strong>Abstract:</strong><br/>
Suppose that $\hat{b}_m\downarrow 0,\ \{\hat{b}_m\}_{m=1}^\infty\notin l^2,$ and $b_n=2^{-\frac{m}{2}}\hat{b}_m$ for all $ n\in(2^m,2^{m+1}].$ In this paper, it is proved that any measurable and almost everywhere finite function $f(x)$ on $[0,1]$ can be corrected on a set of arbitrarily small measure to a bounded measurable function $\widetilde{f}(x)$; so that the nonzero Fourier-Haar coefficients of the corrected function present some subsequence of $\{b_n\}$, and its Fourier-Haar series converges uniformly on $[0,1]$.
</p>projecteuclid.org/euclid.aot/1528444822_20180919220102Wed, 19 Sep 2018 22:01 EDTA Banach algebra with its applications over paths of bounded variationhttps://projecteuclid.org/euclid.aot/1528444823<strong>Dong Hyun Cho</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 4, 794--806.</p><p><strong>Abstract:</strong><br/>
Let $C[0,T]$ denote the space of continuous real-valued functions on $[0,T]$. In this paper we introduce two Banach algebras: one of them is defined on $C[0,T]$ and the other is a space of equivalence classes of measures over paths of bounded variation on $[0,T]$. We establish an isometric isomorphism between them and evaluate analytic Feynman integrals of the functions in the Banach algebras, which play significant roles in the Feynman integration theories and quantum mechanics.
</p>projecteuclid.org/euclid.aot/1528444823_20180919220102Wed, 19 Sep 2018 22:01 EDTOn tensors of factorizable quantum channels with the completely depolarizing channelhttps://projecteuclid.org/euclid.aot/1528444824<strong>Yuki Ueda</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 4, 807--815.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain results for factorizability of quantum channels. Firstly, we prove that if a tensor $T\otimes S_k$ of a quantum channel $T$ on $M_n(\mathbb{C})$ with the completely depolarizing channel $S_k$ is written as a convex combination of automorphisms on the matrix algebra $M_n(\mathbb{C})\otimes M_k(\mathbb{C})$ with rational coefficients, then the quantum channel $T$ has an exact factorization through some matrix algebra with the normalized trace. Next, we prove that if a quantum channel has an exact factorization through a finite dimensional von Neumann algebra with a convex combination of normal faithful tracial states with rational coefficients, then it also has an exact factorization through some matrix algebra with the normalized trace.
</p>projecteuclid.org/euclid.aot/1528444824_20180919220102Wed, 19 Sep 2018 22:01 EDTOn an elasto-acoustic transmission problem in anisotropic, inhomogeneous mediahttps://projecteuclid.org/euclid.aot/1530928840<strong>Rainer Picard</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 4, 816--828.</p><p><strong>Abstract:</strong><br/>
We consider a coupled system describing the interaction between acoustic and elastic regions, where the coupling occurs not via material properties but through an interaction on an interface separating the two regimes. Evolutionary well-posedness in the sense of Hadamard well-posedness supplemented by causal dependence is shown for a natural choice of generalized interface conditions. The results are obtained in a real Hilbert space setting incurring no regularity constraints on the boundary and almost none on the interface of the underlying regions.
</p>projecteuclid.org/euclid.aot/1530928840_20180919220102Wed, 19 Sep 2018 22:01 EDTCompact and “compact” operators on standard Hilbert modules over $C^*$-algebrashttps://projecteuclid.org/euclid.aot/1530928843<strong>Zlatko Lazović</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 4, 829--836.</p><p><strong>Abstract:</strong><br/>
We construct a topology on the standard Hilbert module $H_{\mathcal{A}}$ over a unital $C^*$-algebra and topology on $H_ \mathcal {A} ^{＃}$ (the extension of the module $H_{\mathcal{A}}$ by the algebra $\mathcal{A}^{**}$) such that any “compact” operator (i.e. any operator in the norm closure of the linear span of the operators of the form $z\mapsto x \langle y,z \rangle$, $x,y\in H_{\mathcal{A}}$ (or $x,y \in H_ \mathcal {A} ^{＃}$)) maps bounded sets into totally bounded sets.
</p>projecteuclid.org/euclid.aot/1530928843_20180919220102Wed, 19 Sep 2018 22:01 EDTRegular spectrum of elements in topological algebrashttps://projecteuclid.org/euclid.aot/1532656919<strong>Mati Abel</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 4, 837--854.</p><p><strong>Abstract:</strong><br/>
Main properties of the regular (or extended) spectrum of elements in topological algebras (introduced by L. Waelbroeck and G. R. Allan for unital locally convex algebras) are presented. Descriptions of the relationship between the usual spectrum and the regular spectrum of elements in topological algebras with jointly continuous multiplication are given. It is shown that the usual spectrum and the regular spectrum of elements coincide for Hausdorff locally convex Waelbroeck algebras. Main properties of the disolvent map of elements in topological algebras are studied.
</p>projecteuclid.org/euclid.aot/1532656919_20180919220102Wed, 19 Sep 2018 22:01 EDTThe polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operatorshttps://projecteuclid.org/euclid.aot/1532656920<strong>Na Liu</strong>, <strong>Wei Luo</strong>, <strong>Qingxiang Xu</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 4, 855--867.</p><p><strong>Abstract:</strong><br/>
Let $T$ be an adjointable operator between two Hilbert $C^*$-modules, and let $T^*$ be the adjoint operator of $T$. The polar decomposition of $T$ is characterized as $T=U(T^*T)^\frac{1}{2}$ and $\mathcal{R}(U^*)=\overline{\mathcal{R}(T^*)}$, where $U$ is a partial isometry, $\mathcal{R}(U^*)$ and $\overline{\mathcal{R}(T^*)}$ denote the range of $U^*$ and the norm closure of the range of $T^*$, respectively. Based on this new characterization of the polar decomposition, an application to the study of centered operators is carried out.
</p>projecteuclid.org/euclid.aot/1532656920_20180919220102Wed, 19 Sep 2018 22:01 EDTProjections and isolated points of parts of the spectrumhttps://projecteuclid.org/euclid.aot/1532656921<strong>Pietro Aiena</strong>, <strong>Salvatore Triolo</strong>. <p><strong>Source: </strong>Advances in Operator Theory, Volume 3, Number 4, 868--880.</p><p><strong>Abstract:</strong><br/>
In this paper, we relate the existence of certain projections, commuting with a bounded linear operator $T\in L(X)$ acting on Banach space $X$, with the generalized Kato decomposition of $T$. We also relate the existence of these projections with some properties of the quasi-nilpotent part $H_0(T)$ and the analytic core $K(T)$. Further results are given for the isolated points of some parts of the spectrum.
</p>projecteuclid.org/euclid.aot/1532656921_20180919220102Wed, 19 Sep 2018 22:01 EDT