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

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 l(2)(p)(C) (1 <= p < infinity) and show that the numerical range of the backward shift on an infinite-dimensional space 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 l(n)(p) (n >= 2) is the closed unit disc. Finally, we discuss the sequentially essential numerical ranges of operators on a semi-inner-product space.