The notion of an essentially slant Toeplitz operator on the space $L^2$ is introduced and some of the properties of the set ${\rm ESTO}(L^2)$, the set of all essentially slant Toeplitz operators on $L^2$, are investigated. In particular the conditions under which the product of two operators in ${\rm ESTO}(L^2)$ is in ${\rm ESTO}(L^2)$ are discussed. The notion is generalized to $k$th-order essentially slant Toeplitz operators.

The Mathieu's series $S(r)$ was considered firstly by E.L. Mathieu in 1890, its alternating variant $\widetilde{S}(r)$ has been recently introduced by Pogany et al. [Appl. Math. Comput. 173 (2006), 69--108], where various bounds have been established for $S, \widetilde{S}$. In this note we obtain new upper bounds over $S(r), \widetilde{S}(r)$ with the help of Hardy--Hilbert double integral inequality.

Let $\mathcal{B}$ be a $\sigma$-unital $C^*$-algebra. We show that every strongly continuous $E_0$-semigroup on the algebra of adjointable operators on a full Hilbert $\mathcal{B}$-module $E$ gives rise to a full continuous product system of correspondences over $\mathcal{B}$. We show that every full continuous product system of correspondences over $\mathcal{B}$ arises in that way. If the product system is countably generated, then $E$ can be chosen countably generated, and if $E$ is countably generated, then so is the product system. We show that under these countability hypotheses there is a one-to-one correspondence between $E_0$-semigroups up to stable cocycle conjugacy and continuous product systems up to isomorphism. This generalizes the results for unital $\mathcal{B}$ to the $\sigma$-unital case.

Non-negative definite Lyapunov functionals are employed to obtain sufficient conditions that guarantee boundedness of solutions of system of functional differential equations with unbounded terms. The theory is illustrated with several examples regarding Volterra integro-differential equations.

The aim of this paper is to obtain some new characterizations of Carleson type measure for holomorphic Triebel--Lizorkin spaces and holomorphic Besov type spaces in the unit ball.

We give an alternate proof of the result due to Haydon, Odell and Schlumprecht that subspaces of $L_p$, $p>2$, which are isomorphic to $\ell_2$ contain subspaces which are well isomorphic to $\ell_2$ and well complemented.

We show that a regular von Neumann $Q$-$m$-convex Frechet algebra is of finite dimension. We also show that a regular von Neumann $m$-convex Frechet algebra is a projective limit of finite dimensional algebras. Finally, we prove that a bilateral $Q$-$F$-algebra is a regular von Neumann algebra if and only if it is isomorphic to a finite product of algebras which are also fields.

We shall discuss the matrix geometric mean for the positive definite matrices. The set of all $n\times n$ matrices with a suitable inner product will be a Hilbert space, and the matrix geometric mean can be considered as a path between two positive matrices. In this paper, we shall obtain a matrix geometric mean inequality, and as an application of it, a property of Riemannian metric space is given. We also obtain some examples related to our result.

Let $(\Omega ,\Sigma ,\mu )$ be a complete probability measure space, $E$ be a real separable Banach space, $K$ a nonempty closed convex subset of E. Let $T : \Omega \times K \to K$, such that $\{T_i\}_{i=1}^N$, be N-uniformly $L_i$-Lipschitzian asymptotically hemicontractive random maps of $K$ with $F=\displaystyle\bigcap_{i=1}^N F(T_i)\ne \emptyset$. We construct an explicit iteration scheme and prove neccessary and sufficient conditions for approximating common fixed points of finite family of asymptotically hemicontractive random maps.

The aim of this paper is to prove the stability in the sense of Hyers--Ulam stability of a polynomial equation. More precisely, if $x$ is an approximate solution of the equation $x^n + \alpha x + \beta =0$, then there exists an exact solution of the equation near to $x$.

The investigation of $C^*$-algebras and Hilbert $C^*$-modules with respect to the classical, the weak and the uniform weak Banach--Saks properties is completed giving a full picture, in particular in the non-unital cases. This way some open questions by M. Kusuda and C.-H. Chu are answered. Criteria and structural characterizations are given. In particular, the weak and the uniform weak Banach--Saks property turn out to be invariant under strong Morita equivalence for non-unital $C^*$-algebras.

We show that for $F\in GL(2,\mathbb{C})$, the von Neumann algebra associated to the universal compact quantum group $A_u(F)$ is a free Araki-Woods factor.

Let $K$ be a compact subset of the complex $n$-space and $A(K)$ the algebra of all continuous functions on $K$ which are holomorphic on the interior of $K$. In this paper we show that under some hypotheses on $K$, there exists no linear isometry of finite codimension on $A(K)$. Several compact subsets including the closure of strictly pseudoconvex domain and the product of the closure of plane domains which are bounded by a finite number of disjoint smooth curves satisfy the hypotheses.

We introduce notions of compactness and weak compactness for multilinear maps from a product of normed spaces to a normed space, and prove some general results about these notions. We then consider linear maps $T:A\rightarrow B$ between Banach algebras that are ``close to multiplicative'' in the following senses: the failure of multiplicativity, defined by $S_T(a,b)=T(a)T(b)-T(ab)$ $(a,b\in A)$, is compact [respectively weakly compact]. We call such maps cf-homomorphisms [respectively wcf-homomorphisms]. We also introduce a number of other, related definitions. We state and prove some general theorems about these maps when they are bounded, showing that they form categories and are closed under inversion of mappings and we give a variety of examples. We then turn our attention to commutative $C^*$-algebras and show that the behaviour of the various types of ``close-to-multiplicative'' maps depends on the existence of isolated points in the maximal ideal space. Finally, we look at the splitting of Banach extensions when considered in the category of Banach algebras with bounded cf-homomorphisms [respectively wcf-homomorphisms] as the arrows. This relates to the (weak) compactness of 2-cocycles in the Hochschild-Kamowitz cohomology complex. We prove ``compact'' analogues of a number of established results in the Hochschild-Kamowitz cohomology theory.