scholarly journals Domains of elliptic operators on sets in Wiener space

2020 ◽  
Vol 23 (01) ◽  
pp. 2050004
Author(s):  
Davide Addona ◽  
Gianluca Cappa ◽  
Simone Ferrari
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Quadratic Form ◽  
Gaussian Measure ◽  
Adjoint Operator ◽  
The Self ◽  
Affine Function ◽  
Surface Measure ◽  
Regular Convex

Let [Formula: see text] be a separable Banach space endowed with a non-degenerate centered Gaussian measure [Formula: see text]. The associated Cameron–Martin space is denoted by [Formula: see text]. Consider two sufficiently regular convex functions [Formula: see text] and [Formula: see text]. We let [Formula: see text] and [Formula: see text]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form [Formula: see text] and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator in Hilbert space with [Formula: see text] and on half-spaces, namely if [Formula: see text] and [Formula: see text] is an affine function, then the domain of the operator defined via (0.1) is the space [Formula: see text] where [Formula: see text] is the Feyel–de La Pradelle Hausdorff–Gauss surface measure.

scholarly journals A note on local asymptotic behaviour for Brownian motion in Banach spaces

1979 ◽  
Vol 2 (4) ◽  
pp. 669-676
Author(s):  
Mou-Hsiung Chang
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Brownian Motion ◽  
Gaussian Measure ◽  
Separable Hilbert Space ◽  
Integral Test ◽  
Real Separable Banach Space ◽  
Real Separable Hilbert Space ◽  
Special Case

In this paper we obtain an integral characterization of a two-sided upper function for Brownian motion in a real separable Banach space. This characterization generalizes that of Jain and Taylor [2] whereB=ℝn. The integral test obtained involves the index of a mean zero Gaussian measure on the Banach space, which is due to Kuelbs [3]. The special case that whenBis itself a real separable Hilbert space is also illustrated.

Fractional Self Adjoint Operator Poincaré and Sobolev Type Inequalities

2016 ◽  
Vol 57 (1) ◽  
pp. 5-24 ◽  
Author(s):  
George A. Anastassiou
Keyword(s):  
Hilbert Space ◽  
Adjoint Operator ◽  
The Self ◽  
Sobolev Type

Abstract We present here many fractional self adjoint operator Poincaré and Sobolev type inequalities to various directions. Initially we give several fractional representation formulae in the self adjoint operator sense. Inequalities are based in the self adjoint operator order over a Hilbert space.

scholarly journals Characterizations of commutativity forC*-algebras

1974 ◽  
Vol 15 (2) ◽  
pp. 172-175 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan ◽  
C. M. McGregor
Keyword(s):  
Hilbert Space ◽  
The Self

Letbe aC*-algebra acting on the Hilbert spaceHand letbe the self-adjoint elements of. The following characterization of commutativity is due to I. Kaplansky (see Dixmier [3, p. 58]).

A Generalization of Uniformly Rotund Banach Spaces

1979 ◽  
Vol 31 (3) ◽  
pp. 628-636 ◽  
Author(s):  
Francis Sullivan
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Two Dimensional ◽  
Geometrical Characterization ◽  
Arbitrary Space ◽  
Image Position

Let X be a real Banach space. According to von Neumann's famous geometrical characterization X is a Hilbert space if and only if for all x, y ∈ XThus Hilbert space is distinguished among all real Banach spaces by a certain uniform behavior of the set of all two dimensional subspaces. A related characterization of real Lp spaces can be given in terms of uniform behavior of all two dimensional subspaces and a Boolean algebra of norm-1 projections [16]. For an arbitrary space X, one way of measuring the “uniformity” of the set of two dimensional subspaces is in terms of the real valued modulus of rotundity, i.e. for The space is said to be uniformly rotund if for each 0 we have .

A note on the domain characterization of certain Schrödinger operators with strongly singular potentials

1984 ◽  
Vol 97 ◽  
pp. 125-130 ◽  
Author(s):  
Hubert Kalf
Keyword(s):  
Schrödinger Operators ◽  
Adjoint Operator ◽  
The Self ◽  
Alternative Proof ◽  
Schrodinger Operators ◽  
Singular Potentials ◽  
Strongly Singular Potentials

SynopsisFor β > β0: = 1 −[(n − 2)/2]2 and n ≧ 2, it was recently shown by Simon that the self-adjoint operator associated with −Δ + βr−2 in L2(ℝn) has domain H2(ℝn) ∩D(r−2) the constant β0 being the best possible. An alternative proof of this result is given.

scholarly journals On Abstract Wiener Measure

1972 ◽  
Vol 46 ◽  
pp. 155-160 ◽  
Author(s):  
Balram S. Rajput
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Gaussian Measure ◽  
Separable Hilbert Space ◽  
Reflexive Banach Space ◽  
Wiener Measure ◽  
Real Separable Hilbert Space ◽  
Cylinder Measure

In a recent paper, Sato [6] has shown that for every Gaussian measure n on a real separable or reflexive Banach space (X, ‖ • ‖) there exists a separable closed sub-space X〵 of X such that and is the σ-extension of the canonical Gaussian cylinder measure of a real separable Hilbert space such that the norm is contiunous on and is dense in The main purpose of this note is to prove that ‖ • ‖ x〵 is measurable (and not merely continuous) on .

NEUMARK'S OPERATORS AND SHARP RECONSTRUCTIONS

2006 ◽  
Vol 03 (08) ◽  
pp. 1559-1571 ◽  
Author(s):  
ROBERTO BENEDUCI
Keyword(s):  
Hilbert Space ◽  
Adjoint Operator ◽  
The Self ◽  
The Other ◽  
Projection Formula ◽  
Other Hand ◽  
Random Diffusion

Given a commutative POV measure [Formula: see text] on the Borel σ-algebra of the reals it is possible to construct a PV measure [Formula: see text] (the sharp reconstruction of F) such that the outcomes of the measurement of F can be interpreted as the random diffusion of the outcomes of the measurement of E. On the other hand Neumark's theorem ensures the existence of an extended Hilbert space [Formula: see text] and of a PV measure [Formula: see text] such that [Formula: see text] for every [Formula: see text]. We consider the projection [Formula: see text] of the self-adjoint operator [Formula: see text] corresponding to E+ and the self-adjoint operator A corresponding to the sharp reconstruction E and show the existence of a function G: [0,1] → [0,1] such that [Formula: see text]. An example for which the last relation can be inverted is shown.

scholarly journals Isometric Reflection Vectors and Characterizations of Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-3
Author(s):  
Donghai Ji ◽  
Senlin Wu
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Number Theory ◽  
Recent Result ◽  
Hilbert Spaces ◽  
Unit Sphere ◽  
Substantial Reduction ◽  
Nonempty Interior

A known characterization of Hilbert spaces via isometric reflection vectors is based on the following implication: if the set of isometric reflection vectors in the unit sphereSXof a Banach spaceXhas nonempty interior inSX, thenXis a Hilbert space. Applying a recent result based on well-known theorem of Kronecker from number theory, we improve this by substantial reduction of the set of isometric reflection vectors needed in the hypothesis.

Trace Formulas for Powers of a Sturm-Liouville Operator

1964 ◽  
Vol 16 ◽  
pp. 412-422 ◽  
Author(s):  
Richard C. Gilbert ◽  
Vernon A. Kramer
Keyword(s):  
Hilbert Space ◽  
Differential Operator ◽  
Positive Integer ◽  
Boundary Conditions ◽  
Liouville Operator ◽  
Adjoint Operator ◽  
The Self ◽  
Trace Formulas ◽  
Sturm Liouville

Let H0 be the mth power (m a positive integer) of the self-adjoint operator defined in the Hilbert space L2(0, π) by the differential operator — (d2/dx2) and the boundary conditions u(0) = u(π) = 0. The eigenvalues of H0 are μn = n2m and the corresponding eigenfunctions are ϕn = (2/π)1/2 sin nx, n = 1 , 2 , . . ..Let p be a (2m — 2)-times continuously differentiate real valued function defined over the interval [0, π] satisfying the conditions p(j)(0) = p(j)(π) = 0 for j odd and less than 2m — 4.

Essential Spectra

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Essential Spectrum ◽  
Constant Coefficient ◽  
Differential Operators ◽  
Adjoint Operator ◽  
Closed Operator ◽  
Essential Spectra

In this chapter, various essential spectra are studied. For a closed operator in a Banach space, a number of different sets have been used for the essential spectrum, the sets being identical for a self-adjoint operator in a Hilbert space. As well as the essential spectra, the changes that occur when the operator is perturbed are discussed. Constant-coefficient differential operators are studied in detail.

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