Given a commuting d-tuple T̄ = (T1, . . . , Td) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator DT̄ . Significant attributes of the d-tuple are best expressed in terms of DT̄ , including the Taylor spectrum and the notion of Fredholmness. In fact, all properties of T̄ derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimen...

In the present article, we study discrete spectrum of certain bounded Toeplitz operators with harmonic symbol on a Bergman space. Using methods classical perturbaton theory and recent results by Borichev–Golinskii–Kupin Favorov–Golinskii, obtain quantitative result distribution operator in unbounded (outer) component its Fredholm set.

Approximately fifty percent of Weyl’s theorem fails to transfer from Hilbert space operators to their tensor product. As a biproduct we find that the product of circles in the complex plane is a limaçon. 0. Introduction To say that “Weyl’s theorem holds”, for a bounded operator T ∈ B(X) on a Banach space X , is to assert [2],[4],[5] that 0.1 σ(T ) \ ωess(T ) = π 00 (T ) ≡ iso σ(T )∩π 0 (T ) : t...

In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck operators Lv(x) := A4v(x) + 〈Sx,∇v(x)〉+Df(v?(x))v(x), x ∈ R, d > 2, where v? : R → R is the profile of a rotating wave satisfying v?(x) → v∞ ∈ R as |x| → ∞, the map f : R → R is smooth, the matrix A ∈ R has eigenvalues with positive real parts and commutes with the limit matrix Df(v∞). The matrix S ∈ R is assumed to be...

In this work, an algorithm for finding numerical solutions of linear fractional delay-integro-differential equations (LFDIDEs) variable-order (VO) is introduced. The operational matrices are used as discretization technique based on shifted Chebyshev polynomials (SCPs) the first kind with spectral collocation method. proposed VO-LFDIDEs have multiterms integer, fractional-order derivatives dela...

Integro-differential equations involving Volterra and Fredholm operators (VFIDEs) are used to model many phenomena in science engineering. Nonlocal boundary conditions more effective, some cases necessary, because they accurate measurements of the true state than classical (local) initial conditions. Closed-form solutions always desirable, not only efficient, but also can be valuable benchmarks...

The linearized collision operator of the Boltzmann equation can in a natural way be written as sum positive multiplication operator, frequency, and an integral operator. Compactness for monatomic single species is classical result, while corresponding result mixtures more recently obtained. In this work compactness polyatomic species, where polyatomicity modeled by discrete internal energy vari...

We consider radial complex scaling/perfectly matched layer methods for scalar resonance problems in homogeneous exterior domains. introduce a new abstract framework to analyze the convergence of domain truncations and discretizations. Our theory requires rather minimal assumptions on scaling profile includes affine, smooth, unbounded profiles. report swift technique more technical one approxima...

A bounded operator T on a separable, complex Hilbert space is said to be odd symmetric if IT I = T where I is a real unitary satisfying I = −1 and T t denotes the transpose of T . It is proved that such an operator can always be factorized as T = IAIA with some operator A. This generalizes a result of Hua and Siegel for matrices. As application it is proved that the set of odd symmetric Fredhol...

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis o...