Advances in Operator Theory最新文献

This paper investigates summability principles for multilinear summing operators. The main result presents a novel inclusion theorem for a class of summing operators, which generalizes several classical results. As applications, we derive improved estimates for Hardy–Littlewood inequalities on multilinear forms and prove a Grothendieck-type coincidence result in anisotropic settings.

Let A be a Banach space, (p>1,) and (1/p+1/q=1.) If a sequence (textbf{a}=(a_i)) in A has a finite p-sum, then the operator (Lambda _textbf{a}:ell ^qrightarrow A,) defined by (Lambda _textbf{a}(beta )=sum _{i=1}^infty beta _i a_i,) (beta =(beta _i)in ell ^q,) is compact. We present a characterization of compact operators (Lambda :ell ^qrightarrow A,) and prove that (Lambda ) is compact if and only if (Lambda =Lambda _textbf{a},) for some sequence (textbf{a}=(a_i)) in A with (left{ left( phi (a_i) right) : phi in A^*, Vert phi Vert leqslant 1 right} ) being a totally bounded set in (ell ^p.) For a sequence ((T_i)) of bounded operators on a Hilbert space (mathcal {H},) the corresponding operator ({{varvec{T}}}:ell ^qrightarrow mathbb {B}(mathcal {H}),) defined by ({{varvec{T}}}(beta ) = sum _{i=1}^infty beta _i T_i,) is compact if and only if the set ({langle {{varvec{T}}}x,x rangle :Vert xVert =1}) is a totally bounded subset of (ell ^p,) where (langle {{varvec{T}}}x,x rangle = (langle T_1 x,x rangle , langle T_2 x,x rangle , dotsc ),) for (xin mathcal {H}.) Similar results are established for (p=1) and (p=infty .)

A subclass of weak Dunford–Pettis operators named (hbox {w}^*)DP operators is under investigation. The article studies conditions under which (hbox {w}^*)DP-operators have properties such as (weak) compactness and limitedness, and the relationship of (hbox {w}^*)DP operators with Dunford–Pettis operators. Several further topics related to these operators are investigated.

The linear operators defined on the Lipschitz projective tensor product (X {widehat{boxtimes }}_{pi }E) motivate the study of a distinct class of operators acting on the cartesian product (Xtimes E). These operators, called Lip-linear operators, form a Banach space denoted by (LipL_{0}left( Xtimes E;Fright) .) This space provides an intermediate setting between bilinear operators and two-Lipschitz operators. We establish a natural identification between (LipL_{0}left( Xtimes E;Fright) ) and ({mathcal {L}} (X{widehat{boxtimes }}_{pi }E;F) ,) which also relates it to the space of bilinear operators ({mathcal {B}}left( {mathcal {F}}(X)times E;Fright) ). Furthermore, we extend summability concepts within this category, with a particular focus on integral and dominated (p; q)-summing operators.

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.