Mathematics

We study the vector-valued spectrum M u,??( B ??2 , B ??2 ) which is the set of nonzero algebra homomorphisms from A u ( B ??2 ) (the algebra of uniformly continuous holomorphic functions on B ??2 ) to H ??( B ??2 ) (the algebra of bounded holomorphic functions on B ??2 ). This set is naturally projected onto the closed unit ball of H ??( B ??2 , ??2 ) giving rise to an associated fibering. Extending the classical notion of cluster sets introduced by I. J. Schark (1961) to the vector-valued spectrum we define vector-valued cluster sets. The aim of the article is to look at the relationship between fibers and cluster sets obtaining results regarding the existence of analytic balls into these sets.

The Fock space can be characterized (up to a positive multiplicative factor) as the only Hilbert space of entire functions in which the adjoint of derivation is multiplication by the complex variable. Similarly (and still up to a positive multiplicative factor) the Hardy space is the only space of functions analytic in the open unit disk for which the adjoint of the backward shift operator is the multiplication operator. In the present paper we characterize the Hardy space in term of the adjoint of the differentiation operator. We use reproducing kernel methods, which seem to also give a new characterization of the Fock space.

The main focus of this paper is to define the notion of quasi- (2,β) -Banach space and show some properties in this new space, by help of it and under some natural assumptions, we prove that the fixed point theorem [16, Theorem 2.1] is still valid in the setting of quasi- (2,β) -Banach spaces, this is also an extension of the fixed point result of Brzdęk et al. [12, Theorem 1] in 2 -Banach spaces to quasi- (2,β) -Banach spaces. In the next part, we give a general solution of the radical-type functional equation (1.2). In addition, we study the hyperstability results for these functional equation by applying the aforementioned fixed point theorem, and at the end of this paper we will derive some consequences.

Benedicks theorem for the Weyl Transform states: If the set of points where a function is nonzero is of finite measure, and its Weyl transform is a finite rank operator, then the function is identically zero. A new, more transparent proof of this theorem is given.

Let S be a commutative semigroup, K a quadratically closed commutative field of characteristic different from 2 , G a 2 -cancellative abelian group and H an abelian group uniquely divisible by 2 . The aim of this paper is to determine the general solution f: S 2 ?�K of the d'Alembert type equation: f(x+y,z+w)+f(x+?(y),z+?(w))=2f(x,z)f(y,w),(x,y,z,w?�S) the general solution f: S 2 ?�G of the Jensen type equation: f(x+y,z+w)+f(x+?(y),z+?(w))=2f(x,z),(x,y,z,w?�S) the general solution f: S 2 ?�H of the quadratic type equation quation: f(x+y,z+w)+f(x+?(y),z+?(w))=2f(x,z)+2f(y,w),(x,y,z,w?�S) where ?,?:S?�S are two involutions.

In this note we introduce a new technique to answer an issue posed in [7] concerning geometric properties of the set of non-surjective linear operators. We also extend and improve a related result from the same paper.

Let T denote a positive operator with spectral radius 1 on, say, an L p -space. A classical result in infinite dimensional Perron--Frobenius theory says that, if T is irreducible and power bounded, then its peripheral point spectrum is either empty or a subgroup of the unit circle. In this note we show that the analogous assertion for the entire peripheral spectrum fails. More precisely, for every finite union U of finite subgroups of the unit circle we construct an irreducible stochastic operator on ??1 whose peripheral spectrum equals U . We also give a similar construction for the C 0 -semigroup case.

We study pointwise convergence properties of weakly* converging sequences { u i } i∈N in BV( R n ) . We show that, after passage to a suitable subsequence (not relabeled), we have pointwise convergence u ∗ i (x)→ u ∗ (x) of the precise representatives for all x∈ R n ∖E , where the exceptional set E⊂ R n has on the one hand Hausdorff dimension at most n−1 , and is on the other hand also negligible with respect to the Cantor part of |Du| . Furthermore, we discuss the optimality of these results.

Given a linear map T on a Euclidean Jordan algebra of rank n , we consider the set of all nonnegative vectors q in R n with decreasing components that satisfy the pointwise weak-majorization inequality λ(|T(x)|) ≺ w q∗λ(|x|) , where λ is the eigenvalue map and ∗ denotes the componentwise product in R n . With respect to the weak-majorization ordering, we show the existence of the least vector in this set. When T is a positive map, the least vector is shown to be the join (in the weak-majorization order) of eigenvalue vectors of T(e) and T ∗ (e) , where e is the unit element of the algebra. These results are analogous to the results of Bapat, proved in the setting of the space of all n×n complex matrices with singular value map in place of the eigenvalue map. They also extend two recent results of Tao, Jeong, and Gowda proved for quadratic representations and Schur product induced transformations. As an application, we provide an estimate on the norm of a general linear map relative to spectral norms.

We prove that the only non-Archimedean strictly convex spaces are the zero space and the one-dimensional linear space over Z/3Z , with any of its trivial norms.