Hilbert Space L2 Research Articles

Essential spectrum of non-self-adjoint singular matrix differential operators

The purpose of this paper is to study the essential spectrum of non-self-adjoint singular matrix differential operators in the Hilbert space L2(R)⊕L2(R) induced by matrix differential expressions of the form(0.1)(τ11(⋅,D)τ12(⋅,D)τ21(⋅,D)τ22(⋅,D)), where τ11, τ12, τ21, τ22 are respectively m-th, n-th, k-th and 0 order ordinary differential expressions with m=n+k being even. Under suitable assumptions on their coefficients, we establish an analytic description of the essential spectrum. It turns out that the points of the essential spectrum either have a local origin, which can be traced to points where the ellipticity in the sense of Douglis and Nirenberg breaks down, or they are caused by singularity at infinity.

A modified iterative algorithm for finding a common element in Hilbert space

A modified iterative algorithm for finding a common element in Hilbert space

Spectral radius algebras of WCE operators

In this paper, we investigate the spectral radius al- gebras related to the weighted conditional expectation operators on the Hilbert spaces L2(F). We give a large classes of operators on L2(F) that have the same spectral radius algebra. As a con- sequence we get that the spectral radius algebras of a weighted conditional expectation operator and its Aluthge transformation are equal. Also, we obtain an ideal of the spectral radius algebra related to the rank one operators on the Hilbert space H. Finally we get that the operator T majorizes all closed range elements of the spectral radius algebra of T, when T is a weighted condi- tional expectation operator on L2(F) or a rank one operator on the arbitrary Hilbert space H.

Spectral radius algebras of WCE operators

In this paper, we investigate the spectral radius al- gebras related to the weighted conditional expectation operators on the Hilbert spaces L2(F). We give a large classes of operators on L2(F) that have the same spectral radius algebra. As a con- sequence we get that the spectral radius algebras of a weighted conditional expectation operator and its Aluthge transformation are equal. Also, we obtain an ideal of the spectral radius algebra related to the rank one operators on the Hilbert space H. Finally we get that the operator T majorizes all closed range elements of the spectral radius algebra of T, when T is a weighted condi- tional expectation operator on L2(F) or a rank one operator on the arbitrary Hilbert space H.

Extensions and the weak Calkin algebra of Read’s Banach space admitting discontinuous derivations

Read produced the first example of a Banach space E such that the associated Banach algebra B(E) of bounded operators admits a discontinuous derivation (J. London Math. Soc. 1989). We generalize Read's main theorem about B(E) from which he deduced this conclusion, as well as the key technical lemmas that his proof relied on, by constructing a strongly split-exact sequence {0}→W(E)→B(E)→l2~→{0}, W(E) where W(E) denotes the ideal of weakly compact operators on E, while l2~ is the unitization of the Hilbert space l2, endowed with the zero product.

On unconditional exponential bases in weighted spaces on interval of real axis

In the classical space L2(−π, π) there exists the unconditional basis {ekit} (k is integer). In the work we study the existence of unconditional bases in weighted Hilbert spaces L2(h) of the functions square integrable on an interval (−1, 1) with the weight exp(−h), where h is a convex function. We prove that there exist no unconditional exponential bases in space L2(h) if for some α < 0 (1 − |t|)α = O(eh(t)), t→±1.

Triangularizability of trace-class operators with increasing spectrum

For any measurable set E of a measure space (X,μ), let PE be the (orthogonal) projection on the Hilbert space L2(X,μ) with the range ranPE={f∈L2(X,μ):f=0a.e. onEc} that is called a standard subspace of L2(X,μ). Let T be an operator on L2(X,μ) having increasing spectrum relative to standard compressions, that is, for any measurable sets E and F with E⊆F, the spectrum of the operator PET|ranPE is contained in the spectrum of the operator PFT|ranPF. In 2009, Marcoux, Mastnak and Radjavi asked whether the operator T has a non-trivial invariant standard subspace. They answered this question affirmatively when either the measure space (X,μ) is discrete or the operator T has finite rank. We study this problem in the case of trace-class kernel operators. We also slightly strengthen the above-mentioned result for finite-rank operators.

Approximation of unbounded functional, defined by grades of normal operator, on the class of elements of Hilbert space

We obtain the best approximation of unbounded functional $(A^k x; f)$ on the class $\{ x\in D(A^r) \colon \| A^r x \| \leqslant 1 \}$ by linear bounded functionals for a normal operator $A$ in the Hilbert space $H$ ($k &lt; r$, $f\in H$).

On non-compactly supported p-adic wavelets

For any prime p, we prove the existence of non-compactly supported orthogonal p-adic wavelet bases in the Hilbert space L2(Qp), and construct the first explicit example of such a basis. The reasons are based on a special parametrization of the set of eigen standard Haar vector-functions. It should be noted that all previously known orthogonal p-adic wavelet bases were modifications of the p-adic Haar basis.

On the C *-Algebra Generated by the Bergman Operator, Carleman Second-Order Shift, and Piecewise Continuous Coefficients

We study the C * -algebra generated by the Bergman operator with piecewise continuous coefficients in the Hilbert space L 2 and extended by the Carleman rotation by an angle π. As a result, we obtain an efficient criterion for the operators from the indicated C * -algebra to be Fredholm operators.

An abstract theorem in nonlinear analysis and two applications

We prove abstract multiplicity theorems for a class of nonlinear equations on the Hilbert space L2(Ω). Then we use these theorems to deduce new results for two different types of nonlinear boundary value problems.

Antinormal Weighted Composition Operators

Letl2=L2N,μ, whereNis set of all positive integers andμis the counting measure whoseσ-algebra is the power set ofN. In this paper, we obtain necessary and sufficient conditions for a weighted composition operator to be antinormal on the Hilbert spacel2. We also determine a class of antinormal weighted composition operators on Hardy spaceH2D.

Spectrum of the semi-relativistic Pauli–Fierz model I

A HVZ type theorem for the semi-relativistic Pauli–Fierz Hamiltonian,H=(p⊗1−A)2+M2⊗1+V⊗1+1⊗Hf,M≥0, in quantum electrodynamics is studied. Here H is a self-adjoint operator in Hilbert space L2(Rd)⊗F≅∫Rd⊕Fdx, A=∫Rd⊕A(x)dx is a quantized radiation field and Hf is the free field Hamiltonian defined by the second quantization of a dispersion relation ω:Rd→R. It is emphasized that massless case, M=0, is included. Let E=inf⁡σ(H) be the bottom of the spectrum of H. Suppose that the infimum of ω is m>0. Then it is shown that σess(H)=[E+m,∞). In particular the existence of the ground state of H can be proven.

The Shannon’s mutual information of a multiple antenna time and frequency dependent channel: An ergodic operator approach

Consider a random non-centered multiple antenna radio transmission channel. Assume that the deterministic part of the channel is itself frequency selective and that the random multipath part is represented by an ergodic stationary vector process. In the Hilbert space l2(ℤ), one can associate to this channel a random ergodic self-adjoint operator having a so-called Integrated Density of States (IDS). Shannon’s mutual information per receive antenna of this channel coincides then with the integral of a log function with respect to the IDS. In this paper, it is shown that when the numbers of antennas at the transmitter and at the receiver tend to infinity at the same rate, the mutual information per receive antenna tends to a quantity that can be identified and, in fact, is closely related to that obtained within the random matrix approach [I. Telatar, Eur. Trans. Telecommun. 10, 585 (1999)]. This result can be obtained by analyzing the behavior of the Stieltjes transform of the IDS in the regime of the large numbers of antennas.

On the ideals of a C*-algebra generated by a family of partial isometries and multipliers

The C*-subalgebra of the algebra of all bounded operators on the Hilbert space l2 generated by the multiplier algebra and a family of partial isometries satisfying certain relations is considered. The ideals of the algebra under examination and of its quotient by the algebra of compact operators are studied. It is shown that the quotient algebra can be represented as a direct sum of two principal ideals and has nontrivial center.

An applicable approximation method and its application

In this work, by choosing an orthonormal basis for the Hilbert space L2[0,1], an approximation method for finding approximate solutions of the equation (I+K)x = y is proposed, called Haar wavelet approximation method (HWAM). To prove the applicability of the HWAM, a more general applicability theorem on an approximation method (AM) for an operator equation Ax = y is proved first. As an application, applicability of the HWAM is obtained. Furthermore, four steps to use the HWAM are listed and three numerical examples are given in order to illustrate the effectiveness of the method.

Non-spectrality of self-affine measures on the spatial Sierpinski gasket

Let μM,D be the self-affine measure corresponding to a diagonal matrix M with entries p1,p2,p3∈Z∖{0,±1} and D={0,e1,e2,e3} in the space R3, where e1,e2,e3 are the standard basis of unit column vectors in R3. Such a measure is supported on the spatial Sierpinski gasket. In this paper, we prove the non-spectrality of μM,D. By characterizing the zero set Z(μˆM,D) of the Fourier transform μˆM,D, we obtain that if p1∈2Z and p2,p3∈2Z+1, then μM,D is a non-spectral measure, and there are at most a finite number of orthogonal exponential functions in L2(μM,D). This completely solves the problem on the finiteness or infiniteness of orthogonal exponentials in the Hilbert space L2(μM,D).

Two-Scale Convergence in L 2

We propose a procedure of reducing the notions of weak and strong two-scale convergence to the notions of weak and strong convergence in the Hilbert space L 2 respectively.

A constructive presentation of rigged Hilbert spaces

We construct a rigged Hilbert space for the square integrable functions on the line L2(R) adding to the generators of the Weyl-Heisenberg algebra a new discrete operator, related to the degree of the Hermite polynomials. All together, continuous and discrete operators, constitute the generators of the projective algebra io(2). L2(R) and the vector space of the line R are shown to be isomorphic representations of such an algebra and, as both these representations are irreducible, all operators defined on the rigged Hilbert spaces L2(R) or R are shown to belong to the universal enveloping algebra of io(2). The procedure can be extended to orthogonal and pseudo-orthogonal spaces of arbitrary dimension by tensorialization.Circumventing all formal problems the paper proposes a kind of toy model, well defined from a mathematical point of view, of rigged Hilbert spaces where, in contrast with the Hilbert spaces, operators with different cardinality are allowed.

A necessary and sufficient condition for the finite µ M,D -orthogonality

The self-affine measure µM,D associated with an expanding matrix M ∈ Mn(ℤ) and a finite digit set D ⊂ ℤn is uniquely determined by the self-affine identity with equal weight. The set of orthogonal exponential functions E(Λ) := {e2πi〈λ, x〉 : λ ∈ Λ} in the Hilbert space L2(µM,D) is simply called µM,D-orthogonal exponentials. We consider in this paper the finiteness of µM,D-orthogonality. A necessary and sufficient condition is obtained for the set E(Λ) to be a finite µM,D-orthogonal exponentials. The research here is closely connected with the non-spectrality of self-affine measures.