Hyponormal Operators with Finite Rank Self-Commutators and Quadrature Domains

Abstract

DEDICATED TO PROFESSOR KY FANThis paper studies some classes of pure hyponormal operators with finite rank self-commutators satisfying the condition that their spectra are unions of a finite collection of the closures of quadrature domains. ᮊ

On a Class of Operators of Finite Type

“…Recently, several works [9], [10], [17], [19], [20], [23] have given the natural connection between operator theory and the theory of quadrature domains (cf. [1], [6], [13]).…”

“…Based on α(·), an L(K)-valued measure e(·) on the projection of the boundaries of the quadrature domains in Riemann surfaces is introduced which is an analogue of the L(M )-valued measure e(·) for the pure subnormal operators. In §4, an analytic model for some pure operators of finite type is established, which is the extension of the case dim M = 1 established in [20]. In §5, the mosaic of some pure operators of finite type is studied.…”

“…In §5, the mosaic of some pure operators of finite type is studied. In §6, the relation of two kernels S(·, ·) and E(·, ·) studied in [15] and [17] is also established for a class of operators of finite type. In §7, we study some special case and example.…”

What is a Quadrature Domain?

“…For further results on hyponormal operators and quadrature domains, see [103], [105], [156], [160], [158].…”

“…Other classes of operators which are linked to quadrature domains in different ways are subnormal operators (those operators which can be extended to be normal operators on a larger Hilbert space), see [97], [156], [98], [159], [19], [38], [151] and the Friedrichs operator (essentially the orthogonal projection in L 2 (Ω) of the analytic functions onto the antianalytic ones) [142], [145], [106], [109], [110].…”