Advances in Operator Theory最新文献

The aim of this paper is to investigate the Daugavet equation for a Jordan elementary operator. More precisely, we study the equation

where I stands for the identity operator, A and B are two bounded operators acting on a complex Hilbert space (mathcal {H}), (mathfrak {J}) is a norm ideal of operators on (mathcal {H}), and (U_{mathfrak {J}, A, B}) is the restriction of the Jordan operator (U_{A,B}) to (mathfrak {J}). In the particular case where (mathfrak {J}=mathfrak {C}_{2}(mathcal {H})) is the ideal of Hilbert–Schmidt operators, we give necessary and sufficient conditions under which the above equation holds.

The existence of regular martingale selectors for multivalued supermartingales with unbounded values in a separable Banach space Y is proved. In addition, new convergence results for set-valued supermartingales in the Mosco sense are presented. At the end of this paper, the equivalence between some properties of unbounded set-valued supermartingales and the convergence of these random sets in the Mosco sense is established.

Let (mathcal {A}) be a complex unital Banach algebra. The purpose of this paper is to give a new characterization of generalized n-strong Drazin invertible elements by means of their spectra. Consequently, we address key results in relation with the problem of existence and representations of the generalized n-strong Drazin inverse of the block matrix (x=left( begin{array}{cc}a&{}b c&{}dend{array}right) _{p}) relative to the idempotent p, with a is generalized Drazin invertible such that (a^{d}) is its generalized Drazin inverse in (p mathcal {A}p), under the more general case of the generalized Schur complement (s=d-ca^{d}b) being generalized Drazin invertible.

The diametral dimension, (Delta (E),) and the approximate diametral dimension, (delta (E)) of an element E of a class of nuclear Fréchet spaces, which satisfies ((underline{DN})) and (Omega ) are set theoretically between the respective invariant of power series spaces (Lambda _{1}(varepsilon )) and (Lambda _{infty }(varepsilon )) for some exponent sequence (varepsilon .) Aytuna et al. (Manuscr Math 67:125–142, 1990) proved that E contains a complemented subspace which is isomorphic to (Lambda _{infty }(varepsilon )) provided (Delta (E)= Lambda _{infty }^{prime }(varepsilon ))) and (varepsilon ) is stable. In this article, we consider the other extreme case and we prove that, there exist nuclear Fréchet spaces with the properties ((underline{DN})) and (Omega ,) even regular nuclear Köthe spaces, satisfying (Delta (E)=Lambda _{1}(varepsilon )) such that there is no subspace of E which is isomorphic to (Lambda _{1}(varepsilon ).)

In this article, we establish a multilinear Cotlar-type inequality for the maximal multilinear singular integrals in Dunkl setting whose kernels possess less regularity conditions compared to the multilinear Calderón–Zygmund kernels in spaces of homogeneous type. As applications, we achieve weighted boundedness of maximal multilinear Dunkl–Calderón–Zygmund singular integrals and pointwise convergence of principal value integrals associated with multilinear Dunkl–Calderón–Zygmund kernels.

Twisted convolution is a non-standard convolution which arises while transferring the convolution of the Heisenberg group to the complex plane. Under this operation of twisted convolution, (L^{1}(mathbb {R}^{2n})) turns out to be a non-commutative Banach algebra. Hence the study of (twisted) shift-invariant spaces on (mathbb {R}^{2n}) completely differs from the perspective of the usual shift-invariant spaces on (mathbb {R}^{d}). In this paper, by considering a set of functional data (mathcal {F}={f_{1},ldots ,f_{m}}) in (L^{2}(mathbb {R}^{2n})), we construct a finitely generated twisted shift-invariant space (V^{t}) on (mathbb {R}^{2n}) in such a way that the corresponding system of twisted translates of generators form a Parseval frame sequence and show that it gives the best approximation for a given data, in the sense of least square error. We also find the error of approximation of (mathcal {F}) by (V^{t}). Finally, we illustrate this theory with an example.

Our paper aims to extend fusion frames to Hilbert C(^{*})-modules. We introduce (^*)-fusion frames associated to weighted sequences of closed orthogonally complemented submodules, showcasing similarities to Hilbert space frames. Using Dragan S. Djordjevic’s distance, we define submodule angles and establish a new topology on the set of sequences of closed orthogonally complemented submodules. Relying on this topology, we obtain for our (^*)-fusion frames, some new perturbation results of topological and geometric character.

In this paper, we study free-probabilistic structures of ( C^{*} )-algebras generated by mutually free, multi free random variables followed by the semicircular law in a certain sense. Our main results (i) show that a semicircular element in a ( C^{*} )-probability space (left( A,varphi right) ), and a certain ( * )-isomorphism on A generate countable-infinitely many free random variables followed by the semicircular law, (ii) illustrate that, from mutually free, multi free random variables of (i), the corresponding ( C^{*} )-probability spaces are well-determined, and (iii) characterize not only free-probabilistic structures, but also free-distributional data on the ( C^{*} )-probability spaces of (ii). As application, we study some types of free random variables followed by the circular law, and followed by free Poisson distributions in certain senses.

We consider the quaternion version of the Toeplitz matrix extension problem with prescribed number of negative eigenvalues. The positive semidefinite case is closely related to the Carathéodory–Schur interpolation problem (CSP) in the Schur class (mathcal S_{{mathbb {H}}}) and the Carathéodory class ({mathcal {C}}_{{mathbb {H}}}) of slice-regular functions on the unit quaternionic ball ({mathbb {B}}) that are, respectively, bounded by one in modulus and having positive real part in ({mathbb {B}}). Explicit linear fractional parametrization formulas with free Schur-class parameter for the solution set of the CSP (in the indeterminate case) are given. Carathéodory–Fejér extremal problem and Carathéodory theorem on uniform approximation of a Schur-class function by quaternion finite Blaschke products are also derived. The indefinite version of the Toeplitz extension problem is applied to solve the CSP in the quaternion generalized Schur class. The linear fractional parametrization of the solution set for the indefinite indeterminate problem still exists, but some parameters should be excluded. These excluded parameters and the corresponding “quasi-solutions" are classified and discussed in detail.