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Register a new userDilations for Systems of Imprimitivity acting on Banach Spaces
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Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in general can not be taken as a Hilbert space. However, it can be taken as a Hilbert space for positive operator valued systems of imprimitivity. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.
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We continue the study dilation of linear maps on vector spaces introduced by Bhat, De, and Rakshit. This notion is a variant of vector space dilation introduced by Han, Larson, Liu, and Liu. We derive vector spa
We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis, A. G. cite{AG1} and K{o}nig, W. cite{K} and answers a question of Siskakis, A. G. proposed in cite{AG4}. We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication operator $M_z$.
We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.
We present some properties of orthogonality and relate them with support disjoint and norm inequalities in p Schatten ideals. In addition, we investigate the problem of characterization of norm parallelism for bounded linear operators. We consider the characterization of norm parallelism problem in p Schatten ideals and locally uniformly convex spaces. Later on, we study the case when an operator is norm parallel to the identity operator. Finally, we give some equivalence assertions about the norm parallelism of compact operators. Some applications and generalizations are discussed for certain operators.
In this paper, two related types of dualities are investigated. The first is the duality between left-invertible operators and the second is the duality between Banach spaces of vector-valued analytic functions. We will examine a pair ($mathcal{B},Psi)$ consisting of a reflexive Banach spaces $mathcal{B}$ of vector-valued analytic functions on which a left-invertible multiplication operator acts and an operator-valued holomorphic function $Psi$. We prove that there exist a dual pair ($mathcal{B}^prime,Psi^prime)$ such that the space $mathcal{B}^prime$ is unitarily equivalent to the space $mathcal{B}^*$ and the following intertwining relations hold begin{equation*} mathscr{L} mathcal{U} = mathcal{U}mathscr{M}_z^* quadtext{and}quad mathscr{M}_zmathcal{U} = mathcal{U} mathscr{L}^*, end{equation*} where $mathcal{U}$ is the unitary operator between $mathcal{B}^prime$ and $mathcal{B}^*$. In addition we show that $Psi$ and $Psi^prime$ are connected through the relationbegin{equation*} langle(Psi^prime( bar{z}) e_1) (lambda),e_2rangle= langle e_1,(Psi( bar{ lambda}) e_2)(z)rangle end{equation*} for every $e_1,e_2in E$, $zin varOmega$, $lambdain varOmega^prime$. If a left-invertible operator $T$ satisfies certain conditions, then both $T$ and the Cauchy dual operator $T^prime$ can be modelled as a multiplication operator on reproducing kernel Hilbert spaces of vector-valued analytic functions $mathscr{H}$ and $mathscr{H}^prime$, respectively. We prove that Hilbert space of the dual pair of $(mathscr{H},Psi)$ coincide with $mathscr{H}^prime$, where $Psi$ is a certain operator-valued holomorphic function. Moreover, we characterize when the duality between spaces $mathscr{H}$ and $mathscr{H}^prime$ obtained by identifying them with $mathcal{H}$ is the same as the duality obtained from the Cauchy pairing.
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