Operator Theory, Operator Algebras, and Matrix Theory

Let \({U}_{\infty}{(\mathbb{K})}\) be the locally finite unitriangular group defined over a finite field \({\mathbb{K}}\) with q elements. We define the notion of an indecomposable supercharacter and describe these indecomposable supercharacters in terms of the supercharacters of the finite unitriangular groups \({U}_{n}{(\mathbb{K})}\).

The C^∗-algebra \(\mathfrak{B}\) of bounded linear operators on the space \({L}^{2}(\mathbb{T})\), which is generated by all multiplication operators by piecewise quasicontinuous functions, by the Cauchy singular integral operator and by the range of a unitary representation of a group G of orientation-preserving diffeomorphisms of \({\mathbb{T}}\) onto itself that have the same finite set of fixed points for all \({g} \in {G} \ {\backslash} \ \{e\}\), is studied. A Fredholm symbol calculus for the C^∗-algebra \(\mathfrak{B}\) and a Fredholm criterion for the operators \({B} \in \mathfrak{B}\) are established by using spectral measures and the local-trajectory method for studying C^∗-algebras associated with C^∗-dynamical systems.

The spectral analysis of a family of non-Hermitian operators appearing in quantum physics is our main concern. The properties of such operators are essentially different from those of Hermitian Hamiltonians, namely due to spectral instabilities. We demonstrate that the considered operators and their adjoints can be diagonalized when expressed in terms of certain conveniently constructed operators. We show that their eigenfunctions constitute complete systems, but do not form Riesz bases. Attempts to overcome this difficulty in the quantum mechanical set up are pointed out.

We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a “user’s guide” to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able to read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators (in a sense to be made precise) are invertible. Central to our theoretical results is the concept of a “Fredholm groupoid.” By definition, a Fredholm groupoid is one for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel’s property as the groupoids for which the regular representations are exhaustive. We show that the class of “stratified submersion groupoids” has Exel’s property, where stratified submersion groupoids are defined by gluing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is exploited to yield Fredholm conditions not only in the above-mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.

In the present work, using the concept of statistical e-convergence instead of Pringsheim’s sense for double sequences, we obtain a Korovkin type approximation theorem for double sequences of positive linear operators defined on the space of all real-valued B-continuous functions on a compact subset of the real two-dimensional space. Then, displaying an example, it is shown that our new result is stronger than its classical version.

In this paper we define weighted statistical relative uniform convergence by using weighted density and give a Korovkin-type approximation theorem. Then, we construct an example such that our new approximation result works but its weighted statistical and statistical (and classical) cases do not work. We also compute the rates of weighted statistical relative uniform convergence of sequences of positive linear operators.

In this paper we study simultaneous feedback and output injection on descriptor linear system described by a quadruple of matrices (E,A,B,C). We describe the possible Kronecker invariants of the resulting pencil λE−(A+ BF + KC), when F and K vary, in the case when the pencil corresponding to the system (E,A,B,C) has no infinite elementary divisors of the second, third and fourth type. The solution is constructive and explicit, and is given over algebraically closed fields.

For a tuple \({A} = ({A}_{1}, {A}_{2}, \ldots, {A}_{n})\) of elements in a unital Banach algebra \(\mathcal{B}\), its projective joint spectrum P(A) is the collection of \({z} \in {\mathbb{C}}^{n}\) such that \({A}(z) = {z}_{1}{A}_{1} + {z}_{2}{A}_{2} + \cdots + {z}_{n}{A}_{n}\) is not invertible. It is known that the \(\mathcal{B}\)-valued 1-form \({\omega}_{A}(z) = {A}^{-1}(z){dA}(z)\) contains much topological information about the joint resolvent set P^c(A). This paper studies geometric properties of P^c(A) with respect to Hermitian metrics defined through the \(\mathcal{B}\)-valued fundamental form \({\Omega}_{A} = -{\omega}^{\ast}_{A} \wedge {\omega}_{A}\) and its coupling with faithful states φ on \(\mathcal{B}\), i.e., φ(Ω_A). The connection between the tuple A and the metric is the main subject of this paper. In particular, it shows that the Kählerness of the metric is tied with the commutativity of the tuple, and its completeness is related to the Fuglede–Kadison determinant.

Operator techniques lead to spectral algorithms to compute scaling functions and wavelets associated with multiresolution analyses (MRAs). The spectral algorithms depend on the choice of pairs of suitable orthonormal bases (ONBs). This work presents the spectral algorithms for three different pairs of ONBs: Haar bases, Walsh–Paley bases and trigonometric bases. The Walsh–Paley bases connect wavelet theory and dyadic harmonic analysis. The results for trigonometric bases are the first viable attempt to do a discrete Fourier analysis of the problem.

The nonnegative inverse eigenvalue problem (NIEP) asks which lists of n complex numbers (counting multiplicity) occur as the eigenvalues of some n-by-n entry-wise nonnegative matrix. The NIEP has a long history and is a known hard (perhaps the hardest in matrix analysis?) and sought after problem. Thus, there are many subproblems and relevant results in a variety of directions. We survey most work on the problem and its several variants, with an emphasis on recent results, and include 130 references. The survey is divided into: a) the single eigenvalue problems; b) necessary conditions; c) low-dimensional results; d) sufficient conditions; e) appending 0’s to achieve realizability; f) the graph NIEP’s; g) Perron similarities; and h) the relevance of Jordan structure.

Let α, β be orientation-preserving homeomorphisms of [0,∞] onto itself, which have only two fixed points at 0 and ∞, and whose restrictions to ℝ₊ = (0,∞) are diffeomorphisms, and let U_α, U_β be the corresponding isometric shift operators on the space L^p(ℝ₊) given by \({U}_{\mu}{f} = ({\mu}^\prime)^{1/p}(f\circ\mu)\) for \({\mu} \in \{\alpha, \beta\}\). We prove sufficient conditions for the right and left Fredholmness on L^p(ℝ₊) of singular integral operators of the form \({A}_{+}{P}^{+}_{\gamma} \ {+} \ {A}_{-}{P}^{-}_{\gamma}\) , where \({P}^{\pm}_{\gamma} = ({I} \ {\pm} \ {S}_{\gamma})/2, \ {S}_{\gamma}\) is a weighted Cauchy singular integral operator, \({A}_{+} = \sum\nolimits_{k\in\mathbb{Z}} {a}_{k}{U}^{k}_{\alpha}\) and \({A}_{-} = \sum\nolimits_{k\in\mathbb{Z}} {b}_{k}{U}^{k}_{\beta}\) are operators in the Wiener algebras of functional operators with shifts. We assume that the coefficients a_k, b_k for \({k} \in \mathbb{Z}\) and the derivatives of the shifts \({\alpha}^\prime, {\beta}^\prime\) are bounded continuous functions on ℝ₊ which may have slowly oscillating discontinuities at 0 and ∞.

On the vector Lebesgue space on the unit circle \({L}^{n}_{p} ({p} \in (1, \infty), {n} \in \mathbb{N}\) we consider singular integral operators with a Carleman backward shift of linear fractional type, of the form \({T}_{A,B} = {AP}_{+} + {BP}_{-}\) with A = aI + bU, B = cI + dU, where \({a, b, c, d} \in {L}^{n \times n}, {P}_{\pm} = \frac{1}{2}({I} {\pm} {S})\) are the Cauchy projectors in \({L}^{n}_{p}\) defined componentwise, and U is an involutory shift operator associated with the given Carleman backward shift also defined componentwise. By generalization to the vector case (n > 1) of the previously obtained results for the scalar case (n = 1), it is shown that whenever a certain 2n × 2n matrix function, associated with the original singular integral operator, admits a bounded factorization in \({L}^{2n}_{p}\) the Fredholm characteristics of the paired operator T_A,B can be obtained in terms of that factorization, in particular the dimensions of the kernel and of the cokernel.

The C^∗-algebra \(\mathcal{S}({\mathrm{T}}(C))\) of the finite sections discretization for Toeplitz operators with continuous generating functions is fairly well understood. Since its description in [3], this algebra serves both as a source of inspiration and as an archetypal example of an algebra generated by an discretization procedure. Moreover, it turns out that every separable C^∗-algebra of approximation sequences has a common structure with \(\mathcal{S}({\mathrm{T}}(C))\) after an extension by compact sequences and a suitable fractal restriction. We explain what this statement means and give a proof.

In 1983, the authors introduced a Banach algebra of – as they called them – Toeplitz-like operators. This algebra is defined in an axiomatic way; its elements are distinguished by the existence of four related strong limits. The algebra is in the intersection of Barria and Halmos’ asymptotic Toeplitz operators and of Feintuch’s asymptotic Hankel operators. In the present paper, we start with repeating and extending this approach and introduce Toeplitz and Hankel operators in an abstract and axiomatic manner. In particular, we will see that our abstract Toeplitz operators can be characterized both as shift invariant operators and as compressions. Then we show that the classical Toeplitz and Hankel operators on the spaces \({H}^{p}(\mathbb{T}), {l}^{p}(\mathbb{Z}_{+}), \ \mathrm{and} \ {L}^{p}(\mathbb{R}_{+})\) are concrete realizations of our abstract Toeplitz operators. Finally we generalize some results by Didas on derivations on Toeplitz and Hankel algebras to the axiomatic context.

Algebraic techniques in Numerical Analysis represents the use of Banach and C^∗-algebra theory to help answer questions and solve problems in Numerical Analysis. On the forty-fourth anniversary of the seminal work by Kozak relating stability of an approximation sequence with invertibility in a suitably chosen Banach algebra, the author presents a review of the main issues, history, results and a few of the remaining challenges facing this interdisciplinary area of research, on the frontier between Operator Theory, Banach and C^∗-algebras, and Numerical Analysis.

We give necessary and sufficient conditions for nonlinear control systems of the class C¹ with multi-dimensional control to be linearizable by means of changes of variables of the class C² and additive change of the control of class C¹.

Let u be a Hermitian involution, and e an orthogonal projection, acting on the same Hilbert space \(\mathcal{H}\). We establish the exact formula, in terms of \(||{eue}||\), for the distance from e to the set of all orthogonal projections q from the algebra generated by e, u, and such that quq = 0.