SKEW COMPLEX SYMMETRIC OPERATORS AND WEYL TYPE THEOREMS

Abstract

An operator T 2 L(H) is said to be skew complex symmetric if there exists a conjugation C on H such that T = CT ∗ C. In this pa- per, we study properties of skew complex symmetric operators including spectral connections, Fredholmness, and subspace-hypercyclicity between skew complex symmetric operators and their adjoints. Moreover, we con- sider Weyl type theorems and Browder type theorems for skew complex symmetric operators. Let L(H) be the algebra of all bounded linear operators on a separable complex Hilbert space H and let K(H) be the ideal of all compact operators on H. If T ∈ L(H), we write �(T), �(T), �su(T), �comp(T), �r(T), �c(T), �a(T), �e(T), �le(T), andre(T) for the resolvent set, for the spectrum, the surjective spectrum, the compression spectrum, the residual spectrum, the continuous spectrum, the approximate point spectrum, the essential spectrum, the left essential spectrum, and the right essential spectrum of T, respectively. A conjugation on H is an antilinear operator C : H → H which satisfies hCx,Cyi = hy,xi for all x,y ∈ H and C 2 = I. An antiunitary operator is