Abstract:

The investigation on the numerical range of the complex Volterra operator on Hardy Hilbert space has always been a hot topic for mathematicians, which has not been solved. In this paper, we present the formulas for the numerical radius of an unilateral weighted shift operator with weights $ (h, k, j, b, a, b, a,\cdots ) $, where $ a, b, h, k, j> 0 $. In particular, we apply the above result to calculate the numerical range of the complex Volterra operator on Hardy Hilbert space. These results can not only effectively facilitate further study of the numerical ranges of weighted shift operators with disturbed periodic weights and harmonic weights, but also provide typical examples for the numerical range of bounded linear operators on Hardy Hilbert space.

Key words: Numerical range, Numerical radius, Unilateral weighted shift operator, Complex Volterra operator