Infinite-dimensional Separable Hilbert Space Research Articles

A SIMILARITY BETWEEN THE GROSS LAPLACIAN AND THE LÉVY LAPLACIAN

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

When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator acting on the infinite dimensional separable Hilbert space H ⊕ K of the form M C = A C 0 B . In this paper, it is shown that if A is upper semi-Fredholm of finite ascent and infinite codimension, and if R ( B ) is closed of infinite kernel, then M C is upper semi-Fredholm of finite ascent for some C ∈ B ( K , H ) . In addition, we explore the hypercyclicity and supercyclicity for 2 × 2 upper triangular operator matrices on the Hilbert space.

Additive maps preserving similarity or asymptotic similarity on

Let H be an infinite-dimensional complex separable Hilbert space and denote the algebra of all bounded linear operators acting on H. We show that an additive continuous surjective map Φ on is asymptotic similarity preserving if and only if it is similarity preserving, and in turn, if and only if there exist a scalar c and an invertible bounded linear or conjugate linear operator A on H such that either Φ(T)=cATA −1 for all T or Φ (T)=cAT*A −1 for all T.

AN INFINITE DIMENSIONAL VERSION OF THE SCHUR CONVEXITY PROPERTY AND APPLICATIONS

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.

The spectrum is continuous on the set of quasi- n-hyponormal operators

In this paper it is shown that the spectrum σ, a set-valued function, is continuous when the function is restricted to the set of all ‘quasi- n-hyponormal’ operators acting on an infinite-dimensional separable Hilbert space, where a quasi- n-hyponormal operator is defined to be unitarily equivalent to an n × n upper triangular operator matrix whose diagonal entries are hyponormal operators.

On the Measurability of Additive Functionals

Abstract For an infinite-dimensional separable Hilbert space 𝐻, the problem of measurability of additive functionals 𝑓 : 𝐻 → 𝐑 with respect to various extensions of σ-finite diffused Borel measures on 𝐻 is discussed. It is shown that there exists an everywhere discontinuous additive functional 𝑓 on 𝐻 such that, for any σ-finite diffused Borel measure μ on 𝐻, this 𝑓 can be made measurable with respect to an appropriate extension of μ. Special consideration is given to the case where μ is invariant or quasiinvariant under a subgroup of 𝐻.

On the compact open and finest splitting topologies

In [M.H. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105–145] it is shown that in the set C ( N ω , N ) of all continuous maps of N ω into N, where N is an infinitely countable discrete topological space, the compact-open topology is not the finest splitting topology. Since N ω is consonant (see [S. Dolecki, G.H. Greco, A. Lechicki, When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide? Trans. Amer. Math. Soc. 347 (1995) 2869–2884]) the Isbell topology on C ( N ω , N ) also is not the finest splitting topology. This result is generalized in the present paper proving that it is true also for spaces having the so-called Specific Extension Property. The following spaces have the Specific Extension Property: (a) infinitely countable free unions of non-empty spaces, (b) non-compact Lindelöf zero-dimensional spaces, and (c) metric locally convex linear spaces. In particular, we prove that on the set of all real-valued functions on the (separable infinite dimensional) Hilbert space the compact-open topology does not coincide with the finest splitting topology.

Hypercyclic Conjugate Operators

We prove that for any weighted backward shift B = Bw on an infinite dimensional separable Hilbert space H whose weight sequence w = (wn) satisfies \( \sup_{n} {\left| {w_{1} w_{2} \ldots w_{n} } \right|} = \infty \), the conjugate operator \( C_{B} :S \mapsto BSB^{*} \) is hypercyclic on the space S(H) of self-adjoint operators on H provided with the topology of uniform convergence on compact sets. That is, there exists an \( S \in S(H) \) such that \( \{ C^{n}_{B} (S) = B^{n} SB^{*n}\} _{{n \geq 0}} \) is dense in S(H). We generalize the result to more general conjugate maps \( S \mapsto TST^{*} \), and establish similar results for other operator classes in the algebra B(H) of bounded operators, such as the ideals K(H) and N(H) of compact and nuclear operators respectively.

Extension algebras of Cuntz algebra

In this paper, we construct representatives for all equivalence classes of the unital essential extension algebras of Cuntz algebra by the C *-algebras of compact operators on a separable infinite-dimensional Hilbert space. We also compute their K-groups and semigroups and classify these extension algebras up to isomorphism by their semigroups.

Non-self-adjoint Jacobi matrices with a rank-one imaginary part

We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank-one imaginary part. It is shown that given a set of n not necessarily distinct nonreal numbers in the open upper (lower) half-plane uniquely determines an n × n Jacobi matrix with a rank-one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity. Algorithms of reconstruction for such finite Jacobi matrices are presented. A new model complementing the well-known Livsic triangular model for bounded linear operators with a rank-one imaginary part is obtained. It turns out that the model operator is a non-self-adjoint Jacobi matrix. We show that any bounded, prime, non-self-adjoint linear operator with a rank-one imaginary part acting on some finite-dimensional (respectively separable infinite-dimensional Hilbert space) is unitarily equivalent to a finite (respectively semi-infinite) non-self-adjoint Jacobi matrix. This obtained theorem strengthens a classical result of Stone established for self-adjoint operators with simple spectrum. We establish the non-self-adjoint analogs of the Hochstadt and Gesztesy–Simon uniqueness theorems for finite Jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non-self-adjoint tri-diagonal matrices to the case of mixed eigenvalues, real and nonreal. A unique Jacobi matrix, unitarily equivalent to the operator of integration ( F f ) ( x ) = 2 i ∫ x l f ( t ) d t in the Hilbert space L 2 [ 0 , l ] is found as well as spectral properties of its perturbations and connections with the well-known Bernoulli numbers. We also give the analytic characterization of the Weyl functions of dissipative Jacobi matrices with a rank-one imaginary part.

Floquet operators without singular continuous spectrum

Let U be a unitary operator defined on a infinite-dimensional separable complex Hilbert space H . Assume there exists a self-adjoint operator A on H such that U ∗ A U − A ⩾ c I + K for some positive constant c and compact operator K. Then, assuming the commutators U ∗ A U − A and [ A , U ∗ A U ] admit a bounded extension over H , we prove the spectrum of the operator U has no singular continuous component and only a finite number of eigenvalues of finite multiplicity. We give a localized version of this result and apply it to study the spectrum of the Floquet operator of periodic time-dependent kicked quantum systems.

A SIMPLE PROOF OF THE FUNDAMENTAL THEOREM ABOUT ARVESON SYSTEMS

With every E0-semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E0-semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and allows for a generalization to product systems of Hilbert module (to be published elsewhere).

On bounded solutions to convolution equations

Periodicity of bounded solutions for convolution equations on a separable abelian metric group G is established, and related Liouville type theorems are obtained. A non-constant Borel and bounded harmonic function is constructed for an arbitrary convolution semigroup on any infinite-dimensional separable Hilbert space, generalizing a classical result by Goodman (1973).

Linear Maps Preserving Generalized Invertibility

Let H be an infinite-dimensional complex separable Hilbert space and \( \mathcal{B}(H) \) the algebra of all bounded linear operators on H. Let \( \phi :\mathcal{B}(H) \to \mathcal{B}(H) \) be a bijective continuous unital linear map preserving generalized invertibility in both directions. Then the ideal of all compact operators is invariant under ϕ and the induced linear map on the Calkin algebra is either an automorphism or an antiautomorphism.

The intersection of left (right) spectra of 2 × 2 upper triangular operator matrices

When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator matrix acting on the infinite-dimensional separable Hilbert space H ⊕ K of the form M C = A C 0 B . In this paper, for given A and B, the sets ⋂ C ∈ B l ( K , H ) σ l ( M C ) , ⋂ C ∈ Inv ( K , H ) σ l ( M C ) and ⋃ C ∈ Inv ( K , H ) σ l ( M C ) are determined, where σ l ( T ) , B l ( K , H ) and Inv ( K , H ) denote, respectively, the left spectrum of an operator T, the set of all the left invertible operators and the set of all the invertible operators from K into H .

The topology of the monodromy map of a second order ODE

We consider the following question: given A ∈ SL ( 2 , R ) , which potentials q for the second order Sturm–Liouville problem have A as its Floquet multiplier? More precisely, define the monodromy map μ taking a potential q ∈ L 2 ( [ 0 , 2 π ] ) to μ ( q ) = Φ ˜ ( 2 π ) , the lift to the universal cover G = SL ( 2 , R ) ˜ of SL ( 2 , R ) of the fundamental matrix map Φ : [ 0 , 2 π ] → SL ( 2 , R ) , Φ ( 0 ) = I , Φ ′ ( t ) = ( 0 1 q ( t ) 0 ) Φ ( t ) . Let H be the real infinite-dimensional separable Hilbert space: we present an explicit diffeomorphism Ψ : G 0 × H → H 0 ( [ 0 , 2 π ] ) such that the composition μ ○ Ψ is the projection on the first coordinate, where G 0 is an explicitly given open subset of G diffeomorphic to R 3 . The key ingredient is the correspondence between potentials q and the image in the plane of the first row of Φ, parametrized by polar coordinates, which we call the Kepler transform. As an application among others, let C 1 ⊂ L 2 ( [ 0 , 2 π ] ) be the set of potentials q for which the equation − u ″ + q u = 0 admits a nonzero periodic solution: C 1 is diffeomorphic to the disjoint union of a hyperplane and Cartesian products of the usual cone in R 3 with H .

On the range kernel orthogonality and p-symmetric operators

Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H . For given A ∈ L(H) , we define the derivation δA : L(H) −→ L(H) by δA(X) = AX − XA . In this paper we establish the orthogonality of the range R(δA) and the kernel ker(δA) of a derivation δA induced by a cyclic subnormal operator A , in the usual sense. We give a version of the Putnam Fuglede theorem. We establish a short proof of the principal result of F. Wenying and J. Guoxing in [10]. Relatad results for P-symmetric operators are also given. Mathematics subject classification (2000): 47B47, 47B10, 47A30.

The essential Lagrangian–Grassmannian and the homotopy type of the Fredholm Lagrangian–Grassmannian

Let H be a separable infinite dimensional Hilbert space endowed with a symplectic structure and let L 0 ⊂ H be a Lagrangian subspace. Using the results of [A. Abbondandolo, P. Majer, Infinite dimensional Grassmannians, math.AT/0307192], we show that the Fredholm Lagrangian–Grassmannian F L 0 ( Λ ) has the homotopy type of G c ( L 0 ) , the Grassmannian of all Lagrangian subspaces of H that are compact perturbations of L 0 . It is well known that the latter has the homotopy type of the quotient U ( ∞ ) / O ( ∞ ) . As a corollary, we recover a result by B. Booss-Bavnbek and K. Furutani (see [B. Booss-Bavnbek, K. Furutani, Symplectic functional analysis and spectral invariants, Contemp. Math. 242 (1999) 53–83; K. Furutani, Fredholm–Lagrangian–Grassmannian and the Maslov index, J. Geom. Phys. 51 (2004) 269–331]) that the L 0 -Maslov index is an isomorphism between the fundamental group of F L 0 ( Λ ) and the integers.

The intersection of essential approximate point spectra of operator matrices

When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator acting on the infinite-dimensional separable Hilbert space H ⊕ K of the form M C = ( A C 0 B ) . In this paper, it is shown that there exists some operator C ∈ B ( K , H ) such that M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 if and only if there exists some left invertible operator C ∈ B ( K , H ) such that M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 . A necessary and sufficient condition for M C to be upper semi-Fredholm and ind ( M C ) ⩽ 0 for some C ∈ Inv ( K , H ) is given, where Inv ( K , H ) denotes the set of all the invertible operators of B ( K , H ) . In addition, we give a necessary and sufficient condition for M C to be upper semi-Fredholm and ind ( M C ) ⩽ 0 for all C ∈ Inv ( K , H ) .

Spaces that admit hypercyclic operators with hypercyclic adjoints

A continuous linear operator T : X → X T:X\to X is hypercyclic if there is an x ∈ X x\in X such that the orbit { T n x } n ≥ 0 \{ T^n x\}_{n\geq 0} is dense. A result of H. Salas shows that any infinite-dimensional separable Hilbert space admits a hypercyclic operator whose adjoint is also hypercyclic. It is a natural question to ask for what other spaces X X does L ( X ) \mathcal {L}(X) contain such an operator. We prove that for any infinite-dimensional Banach space X X with a shrinking symmetric basis, such as c 0 c_0 and any ℓ p \ell _p ( 1 > p > ∞ ) (1>p>\infty ) , there is an operator T : X → X T:X \to X , where both T T and T ′ : X ′ → X ′ T’:X’\to X’ are hypercyclic.