Advances in Operator Theory最新文献

Many results for Banach spaces also hold for quasi-Banach spaces. One important such example is results depending on the Baire category theorem (BCT). We use the BCT to explore Lions problem for a quasi-Banach couple ((A_0, A_1).) Lions problem, posed in 1960s, is to prove that different parameters ((theta ,p)) produce different interpolation spaces ((A_0, A_1)_{theta , p}.) We first establish conditions on (A_0) and (A_1) so that interpolation spaces of this couple are strictly intermediate spaces between (A_0+A_1) and (A_0cap A_1.) This result, together with a reiteration theorem, gives a partial solution to Lions problem for quasi-Banach couples. We then apply our interpolation result to (partially) answer a question posed by Pietsch. More precisely, we show that if (pne p^*) the operator ideals ({mathcal {L}}^{(a)}_{p,q}(X,Y),) ({mathcal {L}}^{(a)}_{p^*,q^*}(X,Y)) generated by approximation numbers are distinct. Moreover, for any fixed p,  either all operator ideals ({mathcal {L}}^{(a)}_{p,q}(X,Y)) collapse into a unique space or they are pairwise distinct. We cite counterexamples which show that using interpolation spaces is not appropriate to solve Pietsch’s problem for operator ideals based on general s-numbers. However, the BCT can be used to prove a lethargy result for arbitrary s-numbers which guarantees that, under very minimal conditions on X, Y,  the space ({mathcal {L}}^{(s)}_{p,q}(X,Y)) is strictly embedded into ({mathcal {L}}^{mathcal {A}}(X,Y).)

For an alternative (*)-algebra A under some additional hypothesis, we prove that a map from A into itself is a nonlinear mixed (*)-Jordan type derivation if and only is an additive (*)-derivation. As consequence, some results on the complex octonion algebra, associative (*)-algebras, and (W^*)-factor algebras were obtained.

Let (T:Hrightarrow H) be a bounded operator on Hilbert space H. We say that T has a polygonal type if there exists an open convex polygon (Delta subset {mathbb {D}}), with (overline{Delta }cap {mathbb {T}}ne emptyset ), such that the spectrum (sigma (T)) is included in (overline{Delta }) and the resolvent R(z, T) satisfies an estimate (Vert R(z,T)Vert lesssim max {vert z-xi vert ^{-1},:, xi in overline{Delta }cap {mathbb {T}}}) for (zin overline{mathbb {D}}^c). The class of polygonal type operators (which goes back to De Laubenfels and Franks–McIntosh) contains the class of Ritt operators. Let (T_1,ldots ,T_d) be commuting operators on H, with (dge 3). We prove functional calculus properties of the d-tuple ((T_1,ldots ,T_d)) under various assumptions involving poygonal type. The main ones are the following. (1) If the operator (T_k) is a contraction for all (k=1,ldots ,d) and if (T_1,ldots ,T_{d-2}) have a polygonal type, then ((T_1,ldots ,T_d)) satisfies a generalized von Neumann inequality (Vert phi (T_1,ldots ,T_d)Vert le CVert phi Vert _{infty ,{mathbb {D}}^d}) for polynomials (phi ) in d variables; (2) If (T_k) is polynomially bounded with a polygonal type for all (k=1,ldots ,d), then there exists an invertible operator (S:Hrightarrow H) such that (Vert S^{-1}T_kSVert le 1) for all (k=1,ldots ,d).

We consider the universal additive category derived from the category of equivariant separable (C^*)-algebras by introducing homotopy invariance, stability and split-exactness. We show that morphisms in that category permit a particular simple form, thus obtaining the universal property of (KK^G)-theory for G a locally compact group, or a locally compact groupoid with compact base space, or a countable inverse semigroup as a byproduct.

Let G be a locally compact abelian group with Haar measure and (Phi ) be a Young function. In this paper we characterize the space of bilinear Fourier multipliers as a dual space of a certain Banach algebras for Orlicz spaces.

In this paper we continue the investigation of classes of vector-valued sequences that are represented by Banach operator ideals. By a procedure we mean a correspondence (X mapsto X^{textrm{new}}) that assigns a sequence class (X^{textrm{new}}) built upon a given sequence class X. The general question is whether or not (X^{textrm{new}}) is ideal-representable whenever X is. We address this question for three already studied procedures, namely, (X mapsto X^{textrm{u}}), (X mapsto X^{textrm{dual}}) and (X mapsto X^{textrm{fd}}). Applications of the solutions of these problem will provide new concrete examples of ideal-representable sequence classes.

Our purpose is to identify the graphs that are “fundamental” for the maximum multiplicity problem for Hermitian matrices with a given undirected simple graph. Like paths for trees, these are the special graphs to which the maximum multiplicity problem may be reduced. These are the graphs for which maximum multiplicity implies that all vertices are downers. Examples include cycles and complete graphs, and several more are identified, using the theory developed herein. All the unicyclic graphs that are fundamental, are explicitly identified. We also list those graphs with two edges added to a tree, and their maximum multiplicities, which we have found so far to be fundamental. A formula for maximum multiplicity is given based on fundamental graphs.

Let (mathbb {H}_d:=mathbb {C}^dtimes mathbb {R},) ((din mathbb {N}^*)) be the (2d+1)-dimensional Heisenberg group and we denote by U(d) (the unitary group) the maximal compact connected subgroup of (Aut(mathbb {H}_d),) the group of automorphisms of (mathbb {H}_d.) Let (G_d:=U(d) < imes mathbb {H}_d) be the Heisenberg motion group. In this work, we describe the (C^*)-algebra (C^*(G_d),) of (G_d) in terms of an algebra of operator fields defined over its dual space (widehat{G_d}.) This result generalizes a previous result in Ludwig and Regeiba (Complex Anal Oper Theory 13(8):3943–3978, 2019).

In this paper, we study the approximate minimizing property (AMp) for operators, a localized Bishop-Phelps-Bollobás type property with respect to the minimum norm. Given Banach spaces X and Y we define a new class (mathcal{A}mathcal{M}(X,Y)) of bounded linear operators from X to Y for which the pair (X, Y) satisfies the AMp. We provide a necessary and sufficient condition for non-injective operators from X to Y to be in the class (mathcal{A}mathcal{M}(X,Y)). We also prove that X is finite dimensional if and only if for every Banach space Y, (X, Y) has the AMp for all minimum norm attaining operators from X to Y if and only if for every Banach space Y, (Y, X) has the AMp for all minimum norm attaining operators from Y to X. We also study the AMp with respect to Crawford number called AMp-c for operators.

The aim of the present paper is the investigation of matrix singular integral operators with reflection in Lebesgue spaces on the real line with Muckenhoupt weights. It is proved that these operators are matrix coupled with matrix Toeplitz operators. As a corollary, a criterion for the Fredholmness of such operators with piecewise continuous coefficients is obtained. Singular integral operators with flip and Toeplitz plus Hankel operators are also considered.