Advances in Operator Theory最新文献

In this paper, Roth-type solvability criteria are proposed for the generalized Sylvester equation (AXB+CYD+EZF=G) on finite dimensional spaces and infinite dimensional Hilbert spaces, respectively. Moreover, we give a solvability condition for the operator equation (AXB-CXD = E) by only using one invertible operator.

Recent results on weak characterizations of the Schur property for Banach spaces, based on techniques of bimonotone bases in Banach spaces, are extended to Fréchet—and more general locally convex—spaces by short-cut proofs based on arguments of topological nature.

Every state on the algebra (textrm{M}_n) of complex (ntimes n) matrices restricts to a state on any matrix system. Whereas the restriction to a matrix system is generally not open, we prove that the restriction to every *-subalgebra of (textrm{M}_n) is open. This simplifies topology problems in matrix theory and quantum information theory.

Generalised Morrey (function) spaces enjoyed some interest recently and found applications to PDE. Here we turn our attention to their discrete counterparts. We define generalised Morrey sequence spaces (m_{varphi ,p}=m_{varphi ,p}({mathbb {Z}}^d)). They are natural generalisations of the classical Morrey sequence spaces (m_{u,p}), (0<ple u<infty ), which were studied earlier. We consider some basic features of the spaces as well as embedding properties such as continuity, compactness and strict singularity.

The present paper is devoted to obtain numerical estimations for the equivalences between the Hardy–Littlewood norms of Zygmund’s spaces, (L_{exp }) and (Llog L,) and the Luxemburg norms associated to concrete Young functions that define these spaces. Moreover, for a (finite) measure we compute the equivalence constants between the Hardy–Littlewood norms of Zygmund’s spaces and the norms as associate (Köthe-dual) spaces. It is also proved that, for each (0<r<1,) the quasinorm of the r-convexification (L^r_{exp },) of (L_{exp },) is equivalent to a norm. In the opposite, the quasinorm of the r-convexification (L^rlog L,) of (Llog L,) is not equivalent to a norm. In the atomic case, the r-convexification (L^rlog L) has a separating dual. We analyse the weak compactness of the multiplication operators from (L^{infty }) to (L_{exp }) and from (Llog L) to (L^1.) From the weak compactness of the embeddings follows the reflexivity of certain Lions–Peetre interpolated spaces.

We describe a procedure for extending the inner measure (beta _{_{mathcal {I}}}) associated to an operator ideal (mathcal {I}) to a measure (beta _{_{mathfrak {J}}}) for bounded bilinear operators T. When (mathcal {I}) is injective and closed, we show that (beta _{_{mathfrak {J}}}(T)=0) if and only if (T=RS) for some bounded bilinear operator S and (Rin mathcal {I}). If (mathcal {I}) satisfies the (Sigma _r)-condition, then we establish a convexity inequality for the measure (beta _{_{mathfrak {J}}}) of a bilinear operator interpolated by the real method.

In this paper, we aim to extend the established theory of exponentially m-isometric operators by introducing and exploring the concept of n-quasi-exponentially m-isometric operators. This generalization allows us to investigate a broader class of operators. We provide a comprehensive analysis of various key properties of these operators, which are illustrated through specific matrix representations. An examination of their spectral properties is also provided. The open questions presented at the end pave the way for further research and the continued advancement of the theory of m-isometries and related operators.

In this paper, we discuss k-quasi-m-isometric composition operators and weighted composition operators on directed graphs with one circuit and more than one branching vertex.

On a Banach ideal the set of continuous traces is spanned by the set of positive traces. We extend this result by showing that on an arbitrary ideal the set of all regular traces is the linear span of the set of positive traces.

A linear bounded operator T on a complex Banach space X is said to be power-regular if the sequence ({Vert T^n xVert ^{frac{1}{n}}}_{n=1}^{infty }) is convergent for every (xin X). For unilateral weighted shift S, we give a sufficient condition that S is power-regular. As an application, we construct a class of power-regular operators. Moreover, we show that there exist invertible power-regular bilateral weighted shifts, whose inverses are not power-regular.