Sergei M. Grudsky

Toeplitz operators and the modelling of oscillating discontinuities with the help of Blaschke products

In the present paper we establish theorems about the representation of functions with given asymptotics of the argument in a neighborhood of a discontinuity in the form of a Blaschke product or, more general, in the form of a superposition of a continuous function and a Blaschke product. On this foundation, a theory of normal solvability for Toeplitz operators T(a) on the unit circle whose symbols have oscillating discontinuities is constructed. In particular, the cases of symbols of the form

Infinite Toeplitz Matrices

|\arg a({{e}^{{i\theta }}})|\sim |\theta {{|}^{{ - \lambda }}}(\lambda > 0),|\arg a({{e}^{{i\theta }}})|\sim {{\ln }^{\beta }}|{{\theta }^{{ - 1}}}|(\beta > 1),|\arg a({{e}^{{i\theta }}})|\sim {{e}^{{|\theta {{|}^{{ - \lambda }}}}}}(\lambda > 0)

8. Transient Behavior

are considered.

12. Eigenvectors and Pseudomodes

Let \(\left\{ {{A_n}} \right\}_{n = 1}^\infty\) be a sequence of n × n matrices A n . This sequence is said to be stable if there is an n0 such that the matrices A n are invertible for all n ≥ n0 and

13. Structured Perturbations

\mathop {\sup }\limits_{n \geqslant {n_0}} \left\| {A_n^{ - 1}} \right\| < \infty

10. Extreme Eigenvalues

. Using the convention to put ‖A−1‖ = ∞ if A is not invertible, we can say that \(\left\{ {{A_n}} \right\}_{n = 1}^\infty\) is a stable sequence if and only if