On a complex Hilbert space we consider a symmeric operator with equal defect
numbers and its self-adjoint extension. With such pair of operators we
associate a matrix-valued function (Weyl-Titchmarsh function) analytic in the
upper half-plane with non-negative imaginary part. We show that such function
defines the pair uniquely up to unitary equivalence. Then we investigate
properties of the operators which follow from the assumption that the
Weyl-Titchmarsh function is periodic. Examples include multiplication
operators as well as differential operators of first and second order.
This research was done in collaboration with Prof. E. Tsekanovskii,
Niagara University, NY.