Symmetric Positive Operator Research Articles

Extensions of positive symmetric operators and Krein's uniqueness criteria

We revise Krein's extension theory of positive symmetric operators. Our approach using factorization through an auxiliary Hilbert space has several advantages: it can be applied to non-densely defined transformations and it works in both real and complex spaces. As an application of the results and the construction we consider positive self-adjoint extensions of the modulus square operator T ∗ T of a densely defined linear transformation T and bounded self-adjoint extensions of a symmetric operator. Krein's results on the uniqueness of positive (respectively, norm preserving) self-adjoint extensions are also revised.

Extensions of dissipative operators with closable imaginary part

Given a dissipative operator \(A\) on a complex Hilbert space \(\mathcal{H}\) such that the quadratic form \(f \mapsto \text{Im}\langle f, Af \rangle\) is closable, we give a necessary and sufficient condition for an extension of \(A\) to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.

A Birman–Kreĭn–Vishik–Grubb Theory for Sectorial Operators

We consider densely defined sectorial operators $$A_\pm $$ that can be written in the form $$A_\pm =\pm iS+V$$ with $$\mathcal {D}(A_\pm )=\mathcal {D}(S)=\mathcal {D}(V)$$ , where both S and $$V\ge \varepsilon >0$$ are assumed to be symmetric. We develop an analog to the Birmin–Kreĭn–Vishik–Grubb (BKVG) theory of selfadjoint extensions of a given strictly positive symmetric operator, where we will construct all maximally accretive extensions $$A_D$$ of $$A_+$$ with the property that $$\overline{A_+}\subset A_D\subset A_-^*$$ . Here, D is an auxiliary operator from $$\ker (A_-^*)$$ to $$\ker (A_+^*)$$ that parametrizes the different extensions $$A_D$$ . After this, we will give a criterion for when the quadratic form $$\psi \mapsto {\text{ Re }}\langle \psi ,A_D\psi \rangle $$ is closable and show that the selfadjoint operator $$\widehat{V}$$ that corresponds to the closure is an extension of V. We will show how $$\widehat{V}$$ depends on D, which—using the classical BKVG-theory of selfadjoint extensions—will allow us to define a partial order on the real parts of $$A_D$$ depending on D. All of our results will be presented in a way that emphasizes their connection to the classical BKVG-theory and that shows how the BKVG formulas generalize when considering sectorial operators $$A_\pm =\pm iS+V$$ instead of just a strictly positive symmetric operator V. Applications to second order ordinary differential operators are discussed.

Geometric and operator measures of degeneracy for the set of solutions to the Stieltjes matrix moment problem

The ranks of the limit Weyl intervals are known to serve as the geometric measure of degeneracy of the solution set to a Stieltjes matrix moment problem. This paper puts forward the first operator measure of degeneracy for the solution set to a Stieltjes matrix moment problem in terms of the deficiency vectors of a pair of associated positive symmetric operators. A relationship between the geometric and operator measures of degeneracy for a Stieltjes matrix moment problem is established, from which some corollaries about the Stieltjes matrix moment problem are obtained. Bibliography 19 titles.

Systems of imprimitivity for the clifford group

It is known that if the dimension is a perfect square the Clifford group can be represented by monomial matrices. Another way of expressing this result is to say that when the dimension is a perfect square the standard representation of the Clifford group has a system of imprimitivity consisting of one dimensional subspaces. We generalize this result to the case of an arbitrary dimension. Let k be the square-free part of the dimension. Then we show that the standard representation of the Clifford group has a system of imprimitivity consisting of k-dimensional subspaces. To illustrate the use of this result we apply it to the calculation of SIC-POVMs (symmetric informationally complete positive operator valued measures), constructing exact solutions in dimensions 8 (hand-calculation) as well as 12 and 28 (machine-calculation).

On Fitzpatrick functions of monotone linear operators

Bauschke, Borwein and Wang have shown in [H.H. Bauschke, J.M. Borwein, X. Wang, Fitzpatrick functions and continuous linear monotone operators, Siam J. Optimization, 18 (3) (2007), 789–809] that if FT(⋅,⋅) denotes the Fitzpatrick function of a continuous linear monotone operator T on a separable real Hilbert space H, then FT(x,u)=2qT+∗((u+T∗x)/2), where qA∗ denotes the Fenchel conjugate of the function qA:H→R sending x to 2−1〈x,Ax〉 for an arbitrary continuous positive symmetric operator A∈B(H). Here, T+≔(T+T∗)/2≥0 and A† denotes the Moore–Penrose type inverse of a positive symmetric operator A. The main result of the present paper is a sharpening of the result, achieved by showing that dom(qA∗)=ran(A1/2) and that FT(x,u)=4−1‖(T+1/2)†(u+T∗x)‖2 on dom(qA∗), where dom(f) is the set on which an extended real-valued function f is finite. We will also find a formula for the Fitzpatrick function of a general n-cyclic monotone linear operator in terms of a corresponding ordinary monotone linear operator.

Continuity and Representation of Gaussian Mehler Semigroups

We present sufficient conditions on a Gaussian Mehler semigroup on a reflexive Banach space Eto be induced by a single positive symmetric operator Q \in \(Q \in \mathcal{L}(E^* ,E)\), and give a counterexample which shows that this representation theorem is false in every nonreflexive Banach space with a Schauder basis. We also show that the transition semigroup of a Gaussian Mehler semigroup on a separable Banach space Eacts in a pointwise continuous way, uniformly on compact subsets of E, in the space BUC(E) of bounded uniformly continuous real-valued funtions on E. The transition semigroup is shown to be strongly continuous on BUC(E) if and only if S(t) = Ifor all t⩽ 0

On functions connected with sectorial operators and their extensions

For a closed densely defined sectorial operator using its Friedrichs and von Neumann-Krein m-sectorial extensions, the function of two complex variablesWF(λ,z) which determines the simple part of the operator up to unitary equivalence, is defined and studied. The strong limitQF(λ)=−WF(λ, −∞) is an analog of the Q-function of positive symmetric operator. These functions are used for the description of the resolvents of m-sectorial extensions.

Existence of invariant measures for diffusion processes on a Wiener space

In this paper, we discuss existence of finite invariant measures for diffusion processes with unbounded drift on an abstract Wiener space (B,H,μ). The processes we treat are the ones which have generators formally expressed as — (I/2)D*σD + (6, D ), where D stands for the ^-derivative in the sense of Malliavin calculus, D* the adjoint operator of D with respect to the Wiener measure μ, σ a diffusion coefficient, being a function on B taking values in the space of positive symmetric operators on H, and b a drift coefficient, being an ^-valued function on B. When σ = identity, a given generator has rigorous meaning in L sense under a mild condition on 6, and existence of the associated diffusion and invariant measures was proved by Shigekawa [17] in the case that b is bounded. Partially generalized cases were treated by Vintschger [21] and Zhang [22]. We extend these results to the case where σ is not constant and b is not necessarily bounded but has only some kind of exponential integrability. The diffusion process is constructed by using the theory of Dirichlet forms and the Girsanov transformation, so measures charging no set of zero capacity are allowed to be initial measures. In this paper, however, we restrict our attention to absolutely continuous measures with respect to μ, as in the papers mentioned above. That is, our formulation is as follows: Let {Tt} be the associated Markovian semigroup on L°°(μ). Is there any non-zero p £ L (μ) such that

Square Powers of Singularly Perturbed Operators

AbstractWe use the method of self‐adjoint extensions to define a self‐adjoint operator AT as the singular perturbation of a given self‐adjoint operator A by a singular operator T on a Hilbert space.We also find the structure of a singular operator Q such that the singular perturbation of A2 by Q satisfies (A2)Q = (AT)2. We obtain the explicit form of Q in terms of A and T. A definition of the n‐th power for strictly positive symmetric operators is also given.

Complete continuity of the inverse of a positive symmetric operator.

Let A A be a symmetric positive definite linear transformation defined on a dense subset of a Hilbert space H H , and let H A {H_A} . be the Hilbert space completion of the domain of A A with respect to the inner product ( u , v ) A = ( A u , v ) {(u,v)_A} = (Au,v) . It is shown that the inverse of A A is completely continuous on H A {H_A} if and only if it is completely continuous on H H .

Complete Continuity of the Inverse of a Positive Symmetric Operator

Complete Continuity of the Inverse of a Positive Symmetric Operator