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