Advances in Operator Theory最新文献

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.

Let (mathcal {H}) be a complex Hilbert space with (dim {mathcal {H}}ge 2) and (mathcal {B}(mathcal {H})) be the algebra of all bounded linear operators on (mathcal {H}). For (A, B in mathcal {B}(mathcal {H})), B is called a truncation of A, denoted by (Bprec A), if (B=PAQ) for some projections (P,Qin {mathcal {B}}({mathcal {H}})). And B is called a maximal truncation of A if (Bnot =A) and there is no other truncation C of A such that (Bprec C). We give necessary and sufficient conditions for B to be a maximal truncation of A. Using these characterizations, we determine structures of all bijections preserving truncations of operators in both directions on (mathcal {B}(mathcal {H})).

We provide a characterization when the quaternionic numerical range of a matrix is a closed ball with center 0. The proof makes use of Fejér–Riesz factorization of matrix-valued trigonometric polynomials within the algebra of complex matrices associated with quaternion matrices.

The frame dimension function of a frame ({{mathcal {F}}}= {f_j}_{j=1}^{n}) for an n-dimensional Hilbert space H is the function (d_{{{mathcal {F}}}}(x) = dim {textrm{span}}{ langle x, f_{j}rangle f_{j}: j=1,ldots , N}, 0ne xin H.) It is known that ({{mathcal {F}}}) does phase retrieval for an n-dimensional real Hilbert space H if and only if ({textrm{range}} (d_{{{mathcal {F}}}}) = { n}.) This indicates that the range of the dimension function is one of the good candidates to measure the phase retrievability for an arbitrary frame. In this paper we investigate some structural properties for the range of the dimension function, and examine the connections among different exactness of a frame with respect to its PR-redundance, dimension function and range of the dimension function. A subset (Omega ) of ({1,ldots , n}) containing n is attainable if ({textrm{range}} (d_{{{mathcal {F}}}}) = Omega ) for some frame ({{mathcal {F}}}.) With the help of linearly connected frames, we show that, while not every (Omega ) is attainable, every (integer) interval containing n is always attainable by an n-linearly independent frame. Consequently, ({textrm{range}}(d_{{{mathcal {F}}}})) is an interval for every generic frame for ({mathbb {R}},^n.) Additionally, we also discuss and post some questions related to the connections among ranges of the dimension functions, linearly connected frames and maximal phase retrievable subspaces.

Norm inequalities related to geometric means are discussed by many researchers. Though the operator norm is unitarily invariant one, the numerical radius is not so and unitarily similar. In this paper, we prove some numerical radius inequalities that are related to operator geometric means and spectral geometric ones of real power for positive invertible operators.

We introduce the notion of skeleton with a head in a non-zero real vector space. We prove that skeletons with a head describe order unit spaces geometrically. Next, we prove that the skeleton consists of boundary elements of the positive cone of norm one. We discuss some elementary properties of the skeleton. We also find a condition under which V contains a copy of (ell _{infty }^n) for some (n in {mathbb {N}}) as an order unit subspace.