Kleinian Groups in Hilbert Spaces

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

Browder essential approximate point spectra and hypercyclicity for operator matrices

A long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics is the need to work with unbounded operators. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider the states of a quantum system as nuclear operators, and consider bounded operators as observables. In this case, taking the trace for the product of a nuclear operator (a quantum state) and a bounded operator (a quantum observable) gives the average value of a quantum observable in a fixed state. The existence of such an average for unbounded operators is not guaranteed. If we want to define a space of observables that includes such naturally occurring unbounded operators as the position and momentum operators, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been mathematically correct implemented in the Hilbert space H = L^2(R^N). The so-called space of Schwarz operators, equipped with a system of semi-norms and being a Frechet space, was chosen as the space of states. Schwarz operators appeared to be integral operators whose kernels are functions belonging to the usual Schwarz space. The dual space to the space of Schwarz operators should be considered as the space of quantum observables and it really includes such standard observables as polynomials from the products of the position and momentum operators. In the present work we transfer this approach to the symmetric Fock space H = F(H) over an infinite dimensional separable Hilbert space H. We introduce the space of Schwartz operators in F(H) over an infinite dimensional separable space H and investigate which of the standard operators of quantum white noise belong to the space dual to the space of Schwartz operators.

As is well known, for any operator T on a complex separable Hilbert space, T has the polar decomposition T = U |T | , where U is a partial isometry and |T | is the nonnegative operator (T ∗T ) 1 2 . In 2014, Tian et al. proved that on a complex separable infinite dimensional Hilbert space, any operator admits a polar decomposition in a strongly irreducible sense. More precisely, for any operator T and any e > 0, there exists a decomposition T = (U +K)S , where U is a partial isometry, K is a compact operator with ||K|| < e , and S is strongly irreducible. In this paper, we will answer the question for operators on two-dimensional Hilbert spaces.

Algebraic reflexivity of isometry groups and automorphism groups of some operator structures

Contractions with rank one defect operators and truncated CMV matrices

We extend to infinite dimensional separable Hilbert spaces the Schur convexity property of eigenvalues of a symmetric matrix with real entries. Our framework includes both the case of linear, selfadjoint, compact operators, and that of linear selfadjoint operators that can be approximated by operators of finite rank and having a countable family of eigenvalues. The abstract results of the present paper are illustrated by several examples from mechanics or quantum mechanics, including the Sturm–Liouville problem, the Schrödinger equation, and the harmonic oscillator.

A k-nearest neighbor approach for functional regression

It is known that every bounded operator on an infinite dimensional separable Hilbert space \({\mathcal{H}}\) has an invariant subspace if and only if each pair of idempotents on \({\mathcal{H}}\) has a common invariant subspace. We show that the same equivalence holds for operators and pairs of idempotents that are essentially selfadjoint. We also show that each pair of idempotents on \({\mathcal{H}}\) has a common almost-invariant half-space.

The space of contractive C0-semigroups is a Baire space

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

We formulate and prove a generalization of the isoperimetric inequality in the plane. Using this inequality we construct an unitary space—and in consequence—an isomorphic copy of a separable infinite dimensional Hilbert (Sobolev) space, which turns out also a reproducing kernel Hilbert space.

Bayes compound and empirical Bayes estimation of the mean of a Gaussian distribution on a Hilbert space

Let and be infinite-dimensional separable Hilbert spaces and Lat the lattice of all closed subspaces oh . We describe the general form of pairs of bijective maps ϕ, Ψ : Lat → Lat having the property that for every pair U, V ∊ Lat we have . Then we reformulate this theorem as a description of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known structural results for maps on idempotents are easy consequences.

Let be an infinite dimensional separable complex Hilbert space and let , where is the Banach algebra of all bounded linear operators on . In this paper we prove the following results. If is a operator, then 1. is a hypercyclic operator if and only if D and for every hyperinvariant subspace of . 2. If is a pure, then is a countably hypercyclic operator if and only if and for every hyperinvariant subspace of . 3. has a bounded set with dense orbit if and only if for every hyperinvariant subspace of , .