Abstract: In this paper we give some results on single-valuedness of set- valued maps satisfying linear inclusions.

Abstract: 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 + x + = 0, then there exists an exact solution of the equation near to x.

Abstract: Let $\varphi$ be a holomorphic self-map and $g$ a fixed holomorphic function on the unit ball $B$. The boundedness and compactness of the Volterra composition operator $$T_{g,\varphi} f(z)= \int_0^1 f(\varphi(tz)) \Re g(tz)\frac{dt}{t}$$ on the logarithmic Bloch space and little logarithmic Bloch space are studied in this paper.

Abstract: Let A be a Banach algebra and let ' and be continuous homo- morphisms on A. We consider the following module actions on A, a ·x = '(a)x, x ·a = x (a) (a,x 2 A). We denote by A(', ) the above A-module. We call the Banach algebra A, (', )-weakly amenable if every derivation from A into (A(', )) is inner. In this paper among many other things we investigate the relations between weak amenability and (', )-weak amenability ofA. Some conditions can be imposed on A such that the (' 00 , 00 )-weak amenability of A implies the (', )-weak amenability of A.

Abstract: The Mathieu’s series S(r) was considered firstly by E.L. Mathieu in 1890; its alternating variant S(r) has been recently introduced by Pogany et al. [Some families of Mathieu a-series and alternating Mathieu a-series, 2006] where various bounds have been established for S, S. In this note we obtain new upper bounds over S(r), S(r) with the help of Hardy–Hilbert double integral inequality.

Abstract: Let $\varphi$ be a holomorphic self-map of the open unit disk $D$ on the complex plane and $p,\ q>0.$ In this paper, the boundedness and compactness of composition operator $C_{\varphi}$ from generally weighted Bloch space $B^{p}_{\log}$ to $Q^{q}_{\log}$ are investigated.

Abstract: Let B be a -unital C -algebra. We show that every strongly con- tinuous E0-semigroup on the algebra of adjointable operators on a full Hilbert B-module E gives rise to a full continuous product system of correspondences over B. We show that every full continuous product system of correspondences over 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 hypothe- ses there is a one-to-one correspondence between E0-semigroups up to stable cocycle conjugacy and continuous product systems up to isomorphism. This generalizes the results for unital B to the -unital case.

Abstract: 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.

Abstract: The invertibility of Wiener-Hopf plus Hankel operators with es- sentially bounded Fourier symbols is characterized via certain factorization properties of the Fourier symbols. In addition, a Fredholm criterion for these operators is also obtained and the dimensions of the kernel and cokernel are described.

Abstract: We study the action and properties of a dierential operator in the polydisk, extending some classical results from the unit disk. Using so called dyadic decomposition of the polydisk we find precise connections between quazinorms of holomorphic function in the polydisk with quazinorms on the subframe and the unit disk. All our results were previously well-known in the unit disk.

Abstract: By using a certain operator $S^n$, we introduce a class of holomorphic functions $S_n(\beta )$, and obtain some subordination results. We also show that the set $S_n(\beta )$ is convex and obtain some new differential subordinations related to certain integral operators.

Abstract: The notion of an essentially slant Toeplitz operator on the space L 2 is introduced and some of the properties of the set 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 ESTO(L 2 ) is in ESTO(L 2 ) are discussed. The notion is generalized to kth-order essentially slant Toeplitz operators.

Abstract: 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.

Abstract: Non-negative definite Lyapunov functionals are employed to ob- tain sucient conditions that guarantee boundedness of solutions of system of functional dierential equations with unbounded terms. The theory is illus- trated with several examples regarding Volterra integro-dierenti al equations.

Abstract: Let ( , ,µ) be a complete probability measure space, E be a real separable Banach space, K a nonempty closed convex subset of E. Let T : ◊K ! K, such that {Ti} N=1, be N-uniformly Li-Lipschitzian asymptoti- cally hemicontractive random maps of K with F = N \ i=1 F(Ti) 6 ;. We construct

Abstract: We consider second order hyperbolic equations with unbounded operator's coecients possessing time dependent domain of definition in a Hilbert space. Existence and uniqueness of the strong generalized solution are studied. The proofs rely on a generalization of the well known energy in- tegral method. First, we derive a priori estimates for the strong generalized solutions with the help of Yosida operator approximation. Then, using previ- ous results, we show that the range of the operators generated by the posed problem is dense.

Abstract: We shall discuss the matrix geometric mean for the positive defi- nite matrices. The set of all n ◊ 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 geo- metric 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.

Abstract: 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. 1 Department of Mathematics, Niigata University, Niigata 950-2181, Japan. E-mail address: hatori@math.sc.niigata-u.ac.jp 2 Academic Support Center, Kogakuin University, Tokyo 192-0015, Japan. E-mail address: kt13224@ns.kogakuin.ac.jp Date: Received: 19 June 2009; Revised: 18 September 2009; Accepted: 13 October 2009. ∗ Corresponding author. 2000 Mathematics Subject Classification. Primary 46B04; Secondary 32A38, 46J10.

Abstract: 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.

Abstract: Let $X$ be an infinite dimensional complex vector space. We show that a non-constant endomorphism of $X$ has a proper hyperinvariant subspace if and only if its spectrum is non-void. As an application we show that each non-constant continuous endomorphism of the locally convex space $(s)$ of all complex sequences has a proper closed hyperinvariant subspace.

Abstract: If N 2, then there exist finitely many rotations of the sphere S N such that the set of the corresponding rotation operators on L p (S N ) determines the norm topology for 1 < p 1

Abstract: 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.

Abstract: We examine the uniqueness of the bounded structure of semisim- ple and Mackey complete uniformly A-convex algebras. We also consider the particular locally C -case and the uniform one.

Abstract: We study the connection between conjugations of a special kind of dynamical systems, called P-configurations, and solutions to homogeneous Cauchy type functional equations. We find that any two regular P-configurations are conjugate by a homeomorphism, but cannot be conjugate by a dieomor- phism. This leads us to the following conclusion (answering an open question posed by Paneah): there exist continuous nonlinear solutions to the functional equation: f(t) = f t + 1

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

Abstract: A certain subclass of analytic functions in the open unit disc with negative coefficients is introduced. The new class is defined by means of multiplier transformations. By making use of the familiar concept of neighborhoods of analytic function, the author proves coefficient inequalities, distortion theorems and associated inclusion relations for the $(n,\delta)$-neighborhoods of functions belonging to the new class, which satisfy a certain nonhomogeneous Cauchy-Euler differential equation.