Advances in Operator Theory最新文献

Let (M_theta =({mathbb {R}} < imes {mathbb {C}}^2, underset{theta }{cdot }) (theta ) an irrational number), be the Mautner group. We describe the (C^*)-algebra of (M_theta ) as a subalgebra of (C_0({mathbb {C}}^2,{mathcal {B}}(L^{2}({mathbb {R}}))) )

We prove the qualitative uncertainty principle for the continuous modulated shearlet transform on several classes of groups including Abelian groups, compact extensions of Abelian groups and Heisenberg group. As particular cases, one obtains the qualitative uncertainty principles for the Gabor transform, the wavelet transform and the shearlet transform.

Let ({mathbb D}) be the open unit disk in the complex plane. We characterize the boundedness and compactness of the sum of weighted differentiation composition operators

where (nin {mathbb N}_0), (psi _j), (jin overline{0,n}), are holomorphic functions on ({mathbb D}), and (varphi ), a holomorphic self-maps of ({mathbb D}), acting from Bergman spaces with admissible weights to Zygmund type spaces.

On the basis of the boundedness of singular integral operators, we investigate the boundedness of various linear operators acting on two-weighted Herz spaces with three variable exponents. We obtain the extrapolation theorem as well as the boundedness property of bilinear singular operators. First, we are interested in the case where the triangle inequality is available, and then we develop a theory to extend our results in full generality.

For an arbitrary normed space (mathcal {X}) over a field (mathbb {F}in { mathbb {R}, mathbb {C}},) we define the directed graph (Gamma (mathcal {X})) induced by Birkhoff–James orthogonality on the projective space (mathbb P(mathcal {X}),) and also its nonprojective counterpart (Gamma _0(mathcal {X}).) We show that, in finite-dimensional normed spaces, (Gamma (mathcal {X})) carries all the information about the dimension, smooth points, and norm’s maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian (C^*)-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph (Gamma _0({mathcal {R}})) of a (real or complex) Radon plane ({mathcal {R}}) is isomorphic to the graph (Gamma _0(mathbb {F}^2, {Vert cdot Vert }_2)) of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.

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.