Quasi-nilpotency of Generalized Volterra Operators on Sequence Spaces

Abstract

AbstractWe study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted $$\ell ^p$$ ℓ p spaces $$1<p<+\infty $$ 1 < p < + ∞ . Our main result is that when an analytic symbol g is a multiplier for a weighted $$\ell ^p$$ ℓ p space, then the corresponding generalized Volterra operator $$T_g$$ T g is bounded on the same space and quasi-nilpotent, i.e. its spectrum is $$\{0\}.$$ { 0 } . This improves a previous result of A. Limani and B. Malman in the case of sequence spaces. Also combined with known results about multipliers of $$\ell ^p$$ ℓ p spaces we give non trivial examples of bounded quasi-nilpotent generalized Volterra operators on $$\ell ^p$$ ℓ p . We approach the problem by introducing what we call Schur multipliers for lower triangular matrices and we construct a family of Schur multipliers for lower triangular matrices on $$\ell ^p, 1<p<\infty $$ ℓ p , 1 < p < ∞ related to summability kernels. To demonstrate the power of our results we also find a new class of Schur multipliers for Hankel operators on $$\ell ^2 $$ ℓ 2 , extending a result of E. Ricard.