scholarly journals The space of operator valued functions seen as Hilbert H*-module

2020 ◽  
pp. 1-1
Author(s):  
Zlatko Lazovic
Keyword(s):  
Hilbert Space ◽  
Measure Space ◽  
Orthonormal Basis ◽  
Nuclear Operator ◽  
Measurable Functions

Let M be a space of weakly*-measurable functions F : ? ? B(H) on measure space (?,?,?), for which the function F*F is Gel'fand integrable and Gel'fand integral ? ? F*F d? is a nuclear operator on Hilbert space H. We show that M is Hilbert H*-module which contains an orthonormal basis.

scholarly journals PIP-Space Valued Reproducing Pairs of Measurable Functions

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 52 ◽  
Author(s):  
Jean-Pierre Antoine ◽  
Camillo Trapani
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Measure Space ◽  
Abstract Space ◽  
Initial Space ◽  
Measurable Functions ◽  
Continuous Frames

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space Z : = Z ( X , μ ) , where ( X , μ ) is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a pip-space; (iii) Y and Z are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case.

scholarly journals Spaces of measurable functions

2013 ◽  
Vol 11 (7) ◽  
Author(s):  
Piotr Niemiec
Keyword(s):  
Hilbert Space ◽  
Measure Space ◽  
Finite Measure ◽  
Metrizable Space ◽  
Equivalence Classes ◽  
Convergence In Measure ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Completely Metrizable

AbstractFor a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$, µ), the space M µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$-measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.

scholarly journals Isometries of measurable functions

1981 ◽  
Vol 24 (1) ◽  
pp. 13-26 ◽  
Author(s):  
Michael Cambern
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Finite Measure ◽  
Measurable Operator ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Linear Isometries

Let (X, Σ, μ) be a σ-finite measure space and denote by L∞(X, K) the Banach space of essentially bounded, measurable functions F defined on X and taking values in a separable Hilbert space K. In this article a characterization is given of the linear isometries of L∞(X, K) onto itself. It is shown that if T is such an isometry then T is of the form (T(F))(x) = U(x)(φ(F))(x), where φ is a set isomorphism of Σ onto itself, and U is a measurable operator-valued function such that U(x) is almost everywhere an isometry of K onto itself. It is a consequence of the proof given here that every isometry of L∞(X, K) is the adjoint of an isometry of L1(x, K).

Lacunary ideal convergence in measure for sequences of fuzzy valued functions

2021 ◽  
Vol 40 (3) ◽  
pp. 5517-5526
Author(s):  
Ömer Kişi
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Ideal Convergence ◽  
Convergence In Measure ◽  
Ideal Form ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Convergence Of Sequences

We investigate the concepts of pointwise and uniform I θ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ) . We also introduce the concept of I θ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

scholarly journals Existence of a bounded function of the maximal spectral type

1982 ◽  
Vol 2 (3-4) ◽  
pp. 259-261 ◽  
Author(s):  
V. M. Alexeyev
Keyword(s):  
Hilbert Space ◽  
Spectral Type ◽  
Measure Space ◽  
Bounded Function

Let us denote by the Hilbert space of all functions on the measure space Ω, square-integrable with respect to the measure μ.

Biorthogonal Systems in lp-Spaces

1969 ◽  
Vol 21 ◽  
pp. 625-638 ◽  
Author(s):  
R. Keown ◽  
C. Conatser
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Orthonormal Basis ◽  
Natural Generalization ◽  
Standard Basis ◽  
Biorthogonal Systems ◽  
Lp Spaces ◽  
Standard Bases

Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp-spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp-space. The result is interesting for lp-spaces because of the paucity of standard bases in these spaces.Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp-spaces, where is the space lt and the space lswith 1 < r <2 and 2 < s = r/(r – 1).

scholarly journals Spectral Type of One-Parameter Group of Unitary Operators with Transversal Group

1968 ◽  
Vol 32 ◽  
pp. 141-153 ◽  
Author(s):  
Masasi Kowada
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Spectral Type ◽  
Measure Space ◽  
Unitary Operator ◽  
Separable Hilbert Space ◽  
Unitary Operators ◽  
Parameter Group ◽  
Probability Measure Space

It is an important problem to determine the spectral type of automorphisms or flows on a probability measure space. We shall deal with a unitary operator U and a 1-parameter group of unitary operators {Ut} on a separable Hilbert space H, and discuss their spectral types, although U and {Ut} are not necessarily supposed to be derived from an automorphism or a flow respectively.

scholarly journals Classes of operators on vector valued integration spaces

1977 ◽  
Vol 24 (2) ◽  
pp. 129-138 ◽  
Author(s):  
R. J. Fleming ◽  
J. E. Jamison
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Measure Space ◽  
Separable Hilbert Space ◽  
Hermitian Operator ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Image Position ◽  
Vector Valued ◽  
Uniformly Bounded

AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.

SYMPLECTIC SPACE OR ORTHOGONAL SPACE OF n QUBITS

2011 ◽  
Vol 09 (06) ◽  
pp. 1449-1457
Author(s):  
JIAN-WEI XU
Keyword(s):  
Hilbert Space ◽  
Orthogonal Group ◽  
Symplectic Group ◽  
Orthonormal Basis ◽  
Mathematical Structure ◽  
Symplectic Space ◽  
Orthogonal Space ◽  
Spin Flip

In Hilbert space of n qubits, we introduce symplectic space (n odd) or orthogonal space (n even) via the spin-flip operator. Under this mathematical structure we discuss some properties of n qubits, including homomorphically mapping local operations of n qubits into symplectic group or orthogonal group, and proving that the generalized "magic basis" is just the biorthonormal basis (i.e. the orthonormal basis of both Hilbert space and the orthogonal space). Finally, a demonstrated example is given to discuss the application in physics of this mathematical structure.

Involutory *-antiautomorphisms in Toeplitz algebras

1988 ◽  
Vol 103 (3) ◽  
pp. 473-480
Author(s):  
P. J. Stacey
Keyword(s):  
Hilbert Space ◽  
Orthonormal Basis ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Unilateral Shift ◽  
Integral Power

Let H be a separable complex Hilbert space with orthonormal basis {ei: i ∈ ℕ}, let s be the unilateral shift defined by sei = ei+1 for each i and let K be the algebra of compact operators on H. The present paper classifies the involutory *-anti-automorphisms in the C*-algebra C*(sn, K) generated by K and a positive integral power sn of s. It is shown that, up to conjugacy by *-automorphisms, there are two such involutory *-antiautomorphisms when n is even and one when n is odd.

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