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

Abstract

AbstractWe introduce the numerical range of a bounded linear operator on a semi-inner-product space. We compute the numerical ranges of some operators on $\ell _{2}^{p}(\mathbb{C})$ ℓ 2 p ( C ) $(1\le p < \infty )$ ( 1 ≤ p < ∞ ) and show that the numerical range of the backward shift on an infinite-dimensional space $\ell ^{p}$ ℓ 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 $\ell _{n}^{p}$ ℓ n p $(n \ge 2)$ ( n ≥ 2 ) is the closed unit disc. Finally, we discuss the sequentially essential numerical ranges of operators on a semi-inner-product space.