On Some Product of Two Unbounded Self-Adjoint Operators

We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]). The author was supported in part by CNEPRU: B01820070020 (Ministry of Higher Education, Algeria).     

On a Trace Formula for Functions of Noncommuting Operators

Mathematical Notes - The main result of the paper is that the Lifshits-Krein trace formula cannot be generalized to the case of functions of noncommuting self-adjoint operators. To prove this, we...     

Approximation of Unbounded Functions by Linear Positive Operators

A unified class of linear positive operators has been defined. Using these operators some approximation estimates have been obtained for unbounded functions. For particular linear positive operators these results sharpen and improve the earlier estimates due to Fuhua Cheng (J. Approx. Theory, 1984) and Xiehua Sun (J. Approx. Theory, 1988).     

Symmetric Pairs and Self-Adjoint Extensions of Operators,with Applications to Energy Networks

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator A on a Hilbert space \(\mathcal {H}\), by means of a symmetric pair of operators. A symmetric pair is comprised of densely defined operators \(J: \mathcal {H}_1 \rightarrow \mathcal {H}_2\) and \(K: \mathcal {H}_2 \rightarrow \mathcal {H}_1\) which are compatible in a certain sense. With the appropriate definitions of \(\mathcal {H}_1\) and J in terms of A and \(\mathcal {H}\), we show that \((\textit{JJ}^\star )^{-1}\) is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces \(\ell ^2(G)\) and \(\mathcal {H}_{\mathcal {E}}\) (the energy space).     

Stability Properties of Feynman's Operational Calculus for Exponential Functions of Noncommuting Operators

Stability properties of Feynman's operational calculus are addressed in the setting of exponential functions of noncommuting operators. Applications of some of the stability results are presented. In particular, the time-dependent perturbation theory of nonrelativistic quantum mechanics is presented in the setting of the operational calculus and application of the stability results of this paper to the perturbation theory are discussed.     

Self-Adjoint Operators and Cones

Suppose that K is a cone in a real Hilbert space with K = {0},and that A: is a self-adjoint operator which maps K intoitself. If ||A|| is an eigenvalue of A, it is shown that ithas an eigenvector in the cone. As a corollary, it follows thatif ||A||ⁿ is an eigenvalue of Aⁿ, then ||A|| is an eigenvalueof A which has an eigenvector in K. The role of the support-boundaryof K in the simplicity of the principal eigenvalue ||A|| isinvestigated. If H is a separable Hilbert space, it is shownthat ||A|| (A); that is, the spectral radius of A lies in thespectrum of A. When A is compact, we obtain a very elementaryproof of the Krein-Rutman Theorem in the self-adjoint case withoutassuming that K = {0}.     

Strongly Supercommuting Self-Adjoint Operators

We introduce the notion of strong supercommutativity of self-adjoint operators on a -graded Hilbert space and give some basic properties. We clarify that strong supercommutativity is a unification of strong commutativity and strong anticommutativity. We also establish the theory of super quantization. Applications to supersymmetric quantum field theory and a fermion-boson interaction system are discussed.     

Singular Perturbations of Self-Adjoint Operators

Singular relatively compact perturbations of self-adjoint operators are studied. The results obtained are applied to the Schrödinger operator with a singular potential.     

Unbounded Toeplitz Operators

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H ², with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H ² are also studied. In memory of Paul R. Halmos     

一类插值多项式算子与无界函数逼近

将经典“试探函数组”1,x,x^2应用于扩展乘数法,建立了一个判别线性正算子能否改造为逼近任何无界连续函数的充要条件。利用该条件给出了一类变形的插值多项式算子的收敛性定理,得到了具有一般性的结论。     

A Functional Calculus for n-Tuples of Noncommuting Operators

We employ the notion of slice monogenic functions to define a new functional calculus for an n-tuple of not necessarily commuting operators. This calculus is consistent with the Riesz-Dunford calculus for a single operator. Received: October, 2007. Accepted: February, 2008.     

Singular Rank One Perturbations of Self-Adjoint Operators and Krein Theory of Self-Adjoint Extensions

Gesztesy and Simon recently have proven the existence of the strong resolvent limit A_, for A_, = A + (·), where A is a self-adjoint positive operator, being the A-scale). In the present note it is remarked that the operator A_, also appears directly as the Friedrichs extension of the symmetric operator :=A \{f (A)| f,=0\}. It is also shown that Krein's resolvents formula: (A_{_b,}-z)^-1 =(A-z)^-1+ (·, ) _z, with b=b-(1+z) (_z,_-1),_z= (A-z)^-1 defines a self-adjoint operator A_b, for each and b R¹. Moreover it is proven that for any sequence _n which goes to in there exists a sequence _n0 such that A_b, in the strong resolvent sense.     

Unbounded Quasinormal Operators Revisited

Various characterizations of unbounded closed densely defined operators commuting with the spectral measures of their moduli are established. In particular, Kaufman’s definition of an unbounded quasinormal operator is shown to coincide with that given by the third-named author and Szafraniec. Examples demonstrating the sharpness of results are constructed.     

Point Spectrum of Singularly Perturbed Self-Adjoint Operators

We study the inverse spectral problem for the point spectrum of singularly perturbed self-adjoint operators. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 654–658, May, 2005.     

Perturbation Stability of Coherent Riesz Systems under Convolution Operators

We study the orthogonal perturbation of various coherent function systems (Gabor systems, Wilson bases, and wavelets) under convolution operators. This problem is of key relevance in the design of modulation signal sets for digital communication over time-invariant channels. Upper and lower bounds on the orthogonal perturbation are formulated in terms of spectral spread and temporal support of the prototype, and by the approximate design of worst-case convolution kernels. Among the considered bases, the Weyl–Heisenberg structure which generates Gabor systems turns out to be optimal whenever the class of convolution operators satisfies typical practical constraints.     

Operators of Generalized Translation and Hypergroups Constructed from Self-Adjoint Differential Operators

We construct new examples of operators of generalized translation and convolutions in eigenfunctions of certain self-adjoint differential operators. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 659–668, May, 2005.