In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, is a bounded linear operator on the space L2[0,1] of complex-valued square-integrable functions on the interval [0,1]. On the subspace C[0,1] of continuous functions it represents indefinite integration. It is the operator corresponding to the Volterra integral equations.
Definition
The Volterra operator, V, may be defined for a function f ∈ L2[0,1] and a value t ∈ [0,1], as
\( {\displaystyle V(f)(t)=\int _{0}^{t}f(s)\,ds.} \)
Properties
V is a bounded linear operator between Hilbert spaces, with Hermitian adjoint
\( {\displaystyle V^{*}(f)(t)=\int _{t}^{1}f(s)\,ds.} \)
V is a Hilbert–Schmidt operator, hence in particular is compact.[1]
V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ(V) = {0}.[1]
V is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero), but it is not nilpotent.
The operator norm of V is exactly ||V|| = 2⁄π..[1]
References
"Spectrum of Indefinite Integral Operators". Stack Exchange. May 30, 2012.
Further reading
Gohberg, Israel; Krein, M. G. (1970). Theory and Applications of Volterra Operators in Hilbert Space. Providence: American Mathematical Society. ISBN 0-8218-3627-7.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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