scholarly journals The Topological Support of Gauss Measure on Hilbert Space

1970 ◽  
Vol 38 ◽  
pp. 181-183 ◽  
Author(s):  
Kiyosi Ito
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Closed Subset ◽  
Gauss Measure ◽  
Total Measure ◽  
Radon Probability Measure

Let X be a Hilbert space. The topological support of a Radon probability measure P on X is the least closed subset M of X that carries the total measure 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.

Fixed Point Theorems for Lipspchitzian Semigroups

1989 ◽  
Vol 32 (1) ◽  
pp. 90-97 ◽  
Author(s):  
Hajime ishihara
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Closed Subset ◽  
Fixed Point Theorems ◽  
Nonempty Subset ◽  
Continuous Representation ◽  
Semitopological Semigroup ◽  
Lipschitzian Mappings

AbstractLet U be a nonempty subset of a Banach space, S a left reversible semitopological semigroup, a continuous representation of S as lipschitzian mappings on U into itself, that is for each s ∊ S, there exists ks > 0 such that for x, y ∊ U. We first show that if there exists a closed subset C of U such that then S with lim sups has a common fixed point in a Hilbert space. Next, we prove that the theorem is valid in a Banach space E if lim sups

scholarly journals A common unique fixed point theorem for two random operators in Hilbert spaces

2002 ◽  
Vol 32 (3) ◽  
pp. 177-182 ◽  
Author(s):  
Binayak S. Choudhury
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Hilbert Spaces ◽  
Closed Subset ◽  
Separable Hilbert Space ◽  
Unique Fixed Point ◽  
Random Operators ◽  
Nonempty Closed Subset ◽  
Random Fixed Point

We construct a sequence of measurable functions and consider its convergence to the unique common random fixed point of two random operators defined on a nonempty closed subset of a separable Hilbert space. The corresponding result in the nonrandom case is also obtained.

On a Characterisation of Inner Product Spaces

2001 ◽  
Vol 8 (2) ◽  
pp. 231-236
Author(s):  
G. Chelidze
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Normed Space ◽  
Inner Product ◽  
Product Spaces ◽  
Second Moment ◽  
Inner Product Spaces ◽  
Minimum Value ◽  
Positive Weights ◽  
The Mean

Abstract It is well known that for the Hilbert space H the minimum value of the functional F μ (f) = ∫ H ‖f – g‖2 dμ(g), f ∈ H, is achived at the mean of μ for any probability measure μ with strong second moment on H. We show that the validity of this property for measures on a normed space having support at three points with norm 1 and arbitrarily fixed positive weights implies the existence of an inner product that generates the norm.

scholarly journals Approximation by partial isometries

1986 ◽  
Vol 29 (2) ◽  
pp. 255-261 ◽  
Author(s):  
Pei Yuan Wu
Keyword(s):  
Hilbert Space ◽  
Complex Plane ◽  
Closed Subset ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Unitary Operators ◽  
Nonempty Closed Subset ◽  
Partial Isometries ◽  
Bounded Linear ◽  
Normal Operators

Let B(H) be the algebra of bounded linear operators on a complex separable Hilbert space H. The problem of operator approximation is to determine how closely each operator T ∈B(H) can be approximated in the norm by operators in a subset L of B(H). This problem is initiated by P. R. Halmo [3] when heconsidered approximating operators by the positive ones. Since then, this problem has been attacked with various classes L: the class of normal operators whose spectrum is included in a fixed nonempty closed subset of the complex plane [4], the classes of unitary operators [6] and invertible operators [1]. The purpose of this paper is to study the approximation by partial isometries.

scholarly journals NON-UNIFORM HYPERBOLICITY FOR INFINITE DIMENSIONAL COCYCLES

2013 ◽  
Vol 13 (03) ◽  
pp. 1250026 ◽  
Author(s):  
MÁRIO BESSA ◽  
MARIA CARVALHO
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
The Family ◽  
Uniform Hyperbolicity ◽  
Partially Hyperbolic

Let [Formula: see text] be an infinite dimensional Hilbert space, X a compact Hausdorff space and f : X → X a homeomorphism which preserves a Borel ergodic probability measure which is positive on non-empty open sets. We prove that non-uniformly Anosov cocycles are C0-dense in the family of partially hyperbolic cocycles with non-trivial unstable bundles.

Grüss-Landau inequalities for elementary operators and inner product type transformers in Q and Q* norm ideals of compact operators

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2447-2455
Author(s):  
Milan Lazarevic
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Compact Operators ◽  
Product Type ◽  
Inner Product ◽  
Operator Inequality ◽  
Space Operator ◽  
Bounded Operators ◽  
Normal Operators ◽  
Elementary Operators

For a probability measure ? on ? and square integrable (Hilbert space) operator valued functions {A*t}t??, {Bt}t??, we prove Gr?ss-Landau type operator inequality for inner product type transformers |?? AtXBtd?(t)- ?? At d?(t)X ?? Btd?(t)|2? ? ||??AtA*td?(t)- |??A*td?(t)|2||? (?? B*tX*XBtd?(t)- |X?? Btd?(t)|2)?, for all X ? B(H) and for all ? ? [0,1]. Let p ? 2, ? to be a symmetrically norming (s.n.) function, ? (p) to be its p-modification, ? (p)* is a s.n. function adjoint to ?(p) and ||?||?(p)* to be a norm on its associated ideal C?(p)*(H) of compact operators. If X ? C?(p)*(H) and {?n}?n=1 is a sequence in (0,1], such that ??n=1 ?n = 1 and ??n=1 ||?n-1/2 An f||2+||?-1/2n B*nf||2 < +? for some families {An}? n=1 and {Bn}? n=1 of bounded operators on Hilbert space H and for all f ? H, then ||?? n=1 ?-1n AnXBn-?? n=1 AnX ?? n=1 Bn||?(p)* ? ||???n=1 ?-1n |An|2-|??n=1 An|2 X ? ??n=1 ?-1n |B*n|2-|??n=1 B*n|2||?(p)+, if at least one of those operator families consists of mutually commuting normal operators. The related Gr?ss-Landau type ||?||?(p) norm inequalities for inner product type transformers are also provided.

scholarly journals Some fixed point theorems on nonlinear contractive mappings in Hilbert space

2020 ◽  
Vol 2020 (1) ◽  
pp. 1-12
Author(s):  
Namana Seshagiri Rao ◽  
Karusala Kalyani
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Closed Subset ◽  
Fixed Point Theorems ◽  
Contractive Mappings ◽  
Contraction Condition ◽  
Positive Integers ◽  
Rational Type ◽  
Type Contraction

AbstractThe purpose of this paper is to establish the existence and uniqueness of a common fixed point for a pair of continuous self mappings over a closed subset of Hilbert space satisfying certain nonlinear rational type contraction condition. Further this result is extended and generalized for some positive integers powers of a pair of continuous self mappings and then is devolved for a sequence of continuous self mappings in the Hilbert space. Our results generalize and extend many well known results in the literature.

scholarly journals Uniform probability distribution over all density matrices

2022 ◽  
Author(s):  
Eddy Keming Chen ◽  
Roderich Tumulka
Keyword(s):  
Hilbert Space ◽  
Probability Distribution ◽  
Probability Measure ◽  
Density Matrix ◽  
Uniform Distribution ◽  
Joint Distribution ◽  
Complex Hilbert Space ◽  
Density Matrices ◽  
Finite Dimensional ◽  
Uniform Probability Distribution

AbstractLet $$\mathscr {H}$$ H be a finite-dimensional complex Hilbert space and $$\mathscr {D}$$ D the set of density matrices on $$\mathscr {H}$$ H , i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure u on $$\mathscr {D}$$ D that can be regarded as the uniform distribution over $$\mathscr {D}$$ D . We propose a measure on $$\mathscr {D}$$ D , argue that it can be so regarded, discuss its properties, and compute the joint distribution of the eigenvalues of a random density matrix distributed according to this measure.

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