Numerical ranges and complex symmetric operators in semi-inner-product spaces

Abstract

We introduce the numerical range of a bounded linear operator on a semi-inner-product space. We compute the numerical ranges of some operators on ell _{2}^{p}(mathbb{C})(1le p < infty ) and show that the numerical range of the backward shift on an infinite-dimensional space ell ^{p} is the open unit disc. We define a conjugation and a complex symmetric operator on a semi-inner-product space and discuss complex symmetry in the dual space. We prove some properties of a generalized adjoint of a complex symmetric operator. We also show that the numerical range of the complex conjugation on ell _{n}^{p}(n ge 2) is the closed unit disc. Finally, we discuss the sequentially essential numerical ranges of operators on a semi-inner-product space.