Advances in Operator Theory最新文献

The program of matrix product states on tensor powers ({mathcal {A}}^{otimes {mathbb {Z}}}) of (C^*)-algebras is carried under the assumption that ({mathcal {A}}) is an arbitrary nuclear C*-algebra. For any shift invariant state (omega ), we demonstrate the existence of an order kernel ideal ({mathcal {K}}_omega ), whose quotient action reduces and factorizes the initial data (({mathcal {A}}^{otimes {mathbb {Z}}}, omega )) to the tuple (({mathcal {A}},{mathcal {B}}_omega = {mathcal {A}}^{otimes {mathbb {N}}^times }/{mathcal {K}}_omega , {mathbb {E}}_omega : text{AA }otimes {mathcal {B}}_omega rightarrow {mathcal {B}}_omega , {bar{omega }}: {mathcal {B}}_omega rightarrow {mathbb {C}})), where ({mathcal {B}}_omega ) is an operator system and ({mathbb {E}}_omega ) and ({bar{omega }}) are unital and completely positive maps. Reciprocally, given a (input) tuple (({mathcal {A}},{mathcal {S}},{mathbb {E}},phi )) that shares similar attributes, we supply an algorithm that produces a shift-invariant state on ({mathcal {A}}^{otimes {mathbb {Z}}}). We give sufficient conditions in which the so constructed states are ergodic and they reduce back to their input data. As examples, we formulate the input data that produces AKLT-type states, this time in the context of infinite dimensional site algebras ({mathcal {A}}), such as the (C^*)-algebras of discrete amenable groups.

The envelope algorithm is used to precisely describe the numerical range of a block Toeplitz operator with 2-by-2 affine symbol in the case where the numerical range of the symbol at each point of the unit circle is a circular disk. In this setting, there is at most one flat portion on the boundary of the numerical range. Necessary and sufficient conditions are given for the flat portion to materialize.

We study the boundedness of multidimensional Hardy operators over (textbf{R}^n) in the framework of variable generalised local and global Morrey spaces with power-type weights, where we admit variable exponents for weights. We find conditions on the domain and target spaces ensuring such boundedness. In case of local spaces, these conditions involved values of variable integrability exponents of the domain and target spaces only at the origin and infinity. Due to the variability of the exponents of weights, the obtained results proved to be different corresponding to two distinct cases, which we called up to borderline and overbordeline case. We also pay special attention to a particular case, when the variable domain and target Morrey spaces are related to each other by Adams-type condition. The proofs are based on certain point-wise estimates for the Hardy operators, which allow, in particular, to get a statement on the boundedness from a local Morrey space to an arbitrary Banach function space with lattice property.

In this paper the approximation properties of the partial sums of trigonometric Fourier series for functions within the generalized variation classes (BV(p(n)uparrow infty ,varphi )) and (BLambda (p(n)uparrow infty ,varphi )) are investigated. The primary goal is to determine if these classes can provide better rates of uniform convergence compared to the classical Lebesgue estimate. The results show that under certain conditions, this classes offer improved convergence rates. Specifically, when the modulus of continuity (omega ) and the sequences p(n) and (varphi (n)) satisfy particular growth conditions, the uniform convergence rate can surpass the classical Lebesgue estimate. The paper also demonstrates that the conditions required for these improved estimates are not mutually exclusive, allowing a wide range of acceptable rates for (omega ). Additionally, a function is constructed within the class (H^omega cap BLambda (p(n) uparrow infty , varphi )) (but not in (BV(p(n) uparrow infty , varphi ))) whose Fourier series converges uniformly, emphasizing the advantage of the (BLambda (p(n) uparrow infty , varphi )) class.

Generalized Cesàro operators (C_t), for (tin [0,1)), are investigated when they act on the disc algebra (A({mathbb {D}})) and on the Hardy spaces (H^p), for (1le p le infty ). We study the continuity, compactness, spectrum and point spectrum of (C_t) as well as their linear dynamics and mean ergodicity on these spaces.

We extended the Riesz type weak factorization to symmetric quasi Hardy spaces associated with semifinite subdiagonal algebras and the Haagerup noncommutative (H^{p})-spaces under certain conditions. We also proved weak version of the Szego type factorization for symmetric quasi Hardy spaces associated with semifinite subdiagonal algebras and the Haagerup noncommutative (H^{p})-spaces associated with subdiagonal algebras, which have the universal factorization property.

It is known that the necessary and sufficient conditions of the boundedness of commutators on Morrey spaces are given by Di Fazio, Ragusa and Shirai. Moreover, according to the result of Cruz-Uribe and Fiorenza in 2003, it is given that the weak-type boundedness of the commutators of the fractional integral operators on the Orlicz spaces as the endpoint estimates. In this paper, we gave the extention to the weak-type boundedness on the Orlicz–Morrey spaces.

In decision making a weight vector is often obtained from a reciprocal matrix A that gives pairwise comparisons among n alternatives. The weight vector should be chosen from among efficient vectors for A. Since the reciprocal matrix is usually not consistent, there is no unique way of obtaining such a vector. It is known that all weighted geometric means of the columns of A are efficient for A. In particular, any column and the standard geometric mean of the columns are efficient, the latter being an often used weight vector. Here we focus on the study of the efficiency of the vectors in the (algebraic) convex hull of the columns of A. This set contains the (right) Perron eigenvector of A,  a classical proposal for the weight vector, and the Perron eigenvector of (AA^{T}) (the right singular vector of A), recently proposed as an alternative. We consider reciprocal matrices A obtained from a consistent matrix C by modifying at most three pairs of reciprocal entries contained in a 4-by-4 principal submatrix of C. For such matrices, we give necessary and sufficient conditions for all vectors in the convex hull of the columns to be efficient. In particular, this generalizes the known sufficient conditions for the efficiency of the Perron vector. Numerical examples comparing the performance of efficient convex combinations of the columns and weighted geometric means of the columns are provided.

We show that the c-numerical range of a non-scalar skew-Hermitian quaternion matrix is convex. In fact, included in our result is that the c-numerical range of a skew-Hermitian matrix is a rotation invariant subset of the quaternions with zero real parts.