Anatoly Grigorievich Baskakov

Spectral analysis of a differential operator with an involution

We use the method of similar operators to perform the asymptotic analysis of the spectrum of a differential operator with an involution. We show that such operators have compact resolvent, and that their large eigenvalues are determined by the values of (the Fourier coefficients) of their potential up to a summable sequence.

Slowly Varying at Infinity Operator Semigroups

In this paper we study the asymptotic behavior of bounded semigroups of linear operators acting in Banach spaces. The obtained results are closely connected with stability conditions for solutions to parabolic equations under unrestricted growth of time. Here we make no usual assumption on the existence of the mean value of the initial function.

Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces

We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator

О состояниях обратимости разностных и дифференциальных операторов

{\mathcal{L} = -{\rm d}/{\rm d}t + A}

Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations

L=-d/dt+A in homogeneous function spaces. We provide sufficient conditions for invertibility of such operators in terms of the spectral properties of the operator A and the semigroup generated by A. We introduce a homogeneous space of functions with absolutely summable spectrum and prove a generalization of the Gearhart–Prüss Theorem for such spaces. We use the results to prove existence and uniqueness of solutions of a certain class of non-linear equations.

Спектральный анализ дифференциальных операторов с неограниченными операторными коэффициентами, разностные отношения и полугруппы разностных отношений@@@Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations

We consider a second-order linear differential equation whose coefficients are bounded operators acting in a complex Banach space. For this equation with a bounded right-hand side, we study the question on the existence of solutions which are bounded on the whole real axis. An asymptotic behavior of solutions is also explored. The research is conducted under condition that the corresponding “algebraic” operator equation has separated roots or provided that an operator placed in front of the first derivative in the equation has a small norm. In the latter case we apply the method of similar operators, i.e., the operator splitting theorem. To obtain the main results we make use of theorems on the similarity transformation of a second order operator matrix to a block-diagonal matrix.

SLANTED MATRICES, BANACH FRAMES, AND SAMPLING

For investigated linear differential operator (equation) with unbounded periodic operator coefficients defined at one of the Banach space of vector functions defined on all real axis difference operator (equation) with constant operator coefficient defined at appropriate Banach space of two-side vector sequences is considered. For differential and difference operators propositions about kernel and co-image dimensions coincidence, simultaneous complementarity of kernels and images, simultaneous reversibility, spectrum interrelation are proved

The method of similar operators in the spectral analysis of non-self-adjoint Dirac operators with non-smooth potentials

In this paper we present a brief account of the use of the spectral theory of slanted matrices in frame and sampling theory. Some abstract results on slanted matrices are also presented.