Quasi-nilpotency of Generalized Volterra Operators on Sequence Spaces

We 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.

Embedding of Dirichlet type spaces into tent spaces and Volterra operators

In this paper, we study the boundedness and compactness of the inclusion mapping from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to tent spaces. Meanwhile, the boundedness, compactness, and essential norm of Volterra integral operators from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to general function spaces are also investigated.

Few Remarks on Quasi Quantum Quadratic Operators on 𝕄2(ℂ)

The present paper deals with a connection between quantum quadratic operators (QQOs) and quasi QQOs on 𝕄2(ℂ). We show that QQOs and quasi QQOs on 𝕄2(ℂ) coincide in the class of Volterra type of operators. To establish this result, we first describe these two kind of operators on the commutative part of 𝕄2(ℂ). Furthermore, in the last section, we introduce a quantum analogue of Volterra operators and provide concrete examples of such kind of operators. It is established that the considered examples also satisfy quasiness condition as well. The obtained results will allow to produce (with explicit conditions) a class of unital, but not trace-preserving positive maps of 𝕄2(ℂ).

Operator equation and related questions of stability of differential equations

The monograph is devoted to the application of methods of functional analysis to the problems of qualitative theory of differential equations. Describes an algorithm to bring the differential boundary value problem to an operator equation. The research of solutions to operator equations of special kind in the spaces polutoratonny with a cone, where the limitations of the elements of these spaces is understood as the comparability them with a fixed scale element of exponential type. Found representations of the solutions of operator equations in the form of contour integrals, theorems of existence and uniqueness of such solutions. The spectral criteria for boundedness of solutions of operator equations and, as a consequence, sufficient spectral features boundedness of solutions of differential and differential-difference equations in Banach space. The results obtained for operator equations with operators and work of Volterra operators, allowed to extend to some systems of partial differential equations known spectral stability criteria for solutions of A. M. Lyapunov and also to generalize theorems on the exponential characteristic. The results of the monograph may be useful in the study of linear mechanical and electrical systems, in problems of diffraction of electromagnetic waves, theory of automatic control, etc. It is intended for researchers, graduate students functional analysis and its applications to operator and differential equations.