Advances in Operator Theory最新文献

In this paper, we introduce a new family of function spaces of Besov-Triebel-Lizorkin type. We present the (varphi )-transform characterization of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev and Franke-Jewarth embeddings. Also, we establish the smooth atomic, molecular and wavelet decomposition of these function spaces. Characterizations by ball means of differences are given. Finally, we investigate a series of examples which play an important role in the study of function spaces of Besov-Triebel-Lizorkin type.

In this paper, we discuss measure-theoretic characterizations for composition operators in some operator classes on (L^{2}(Sigma ))-semi-Hilbertian spaces with respect to positive multiplication operators.

Let X be a Banach space and T be a bounded linear operator on X. Ergodic type theorems investigate the strong convergence of the Cesàro averages given by (M_n(T):=dfrac{1}{n}sum nolimits _{k=1}^{n}T^k). The main aim of this paper is to give an analogue of mean ergodic type theorems by replacing the ordinary convergence with the ideal convergence. We also present an important inequality that immediately leads to a characterization for the set of closure of the range of (I-T). The results presented in this paper extend some existing theorems in the literature.

Let (n>1) and let ({U_{ij}}_{1le i<jle n}) be (natopwithdelims ()2) commuting unitaries on a Hilbert space ({mathcal {H}}). Suppose (U_{ji}:=U^*_{ij}), (1le i<jle n). An n-tuple of power partial isometries ((V_1,...,V_n)) on Hilbert space ({mathcal {H}}) is called ({mathcal {U}}_n)-twisted power partial isometry with respect to ({U_{ij}}_{i<j}) (or simply ({mathcal {U}}_n)-twisted power partial isometry if ({U_{ij}}_{i<j}) is clear from the context) if (V_i^*V_j=U_{ij}V_jV^*_i, ~~ V_iV_j=U_{ji}V_jV_i ~~text {and}~~ V_kU_{ij}=U_{ij}V_k~~(i,j,k=1,2,...,n,~text {and}~ine j).) We prove that each ({mathcal {U}}_n)-twisted power partial isometry admits a Halmos and Wallen (J Math Mech 19:657–663, 1969/1970) type orthogonal decomposition. We provide a concrete model for the decomposition of ({mathcal {U}}_n)-twisted power partial isometries.

In this paper, we study basic properties of compact operators on Hilbert C*-modules over an arbitrary finite dimensional C*-algebra (mathcal {A}). We introduce the notions of invariant and hyperinvariant submodules in the setting of Hilbert C*-modules and prove a Lomonosov type theorem for compact modular operators on such modules. Specifically, we show that every nonzero compact modular operator acting on a Hilbert (mathcal {A})-module admits a proper nonzero hyperinvariant submodule.

In this paper, we introduce the notion of Shilov boundary ideal for a local operator system and investigate some of its properties.

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.