Volterra operator explained

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^[1]

Properties

• V is a bounded linear operator between Hilbert spaces, with kernel form$$Vf(x)\; =\; \backslash int\_0^1\; 1\_\; f(y)\; dy$$ proven by exchanging the integral sign.
• V is a Hilbert–Schmidt operator with norm

, hence in particular is compact.

• Its Hermitian adjoint has kernel form$$V^*(f)(x)\; =\; \backslash int\_^\; f(y)dy\; =\; \backslash int\_0^1\; 1\_\; f(y)dy$$
• The positive-definite integral operator

has kernel form$$Kf(x)\; =\; \backslash int\_0^1\; \backslash min(1-x,\; 1-y)\; f(y)\; dy$$proven by exchanging the integral sign. Similarly, has kernel . They are unitarily equivalent via , so both have the same spectrum.

• The eigenfunctions of

satisfy with solution $$f(x)\; =\; \backslash sin((k+1/2)\; \backslash pi\; x),\; \backslash lambda\; =\; \backslash left(\backslash frac\backslash right)^2$$with .

• The singular values of V are

with .

• The operator norm of V is

.

• V is not trace class.
• V has no eigenvalues and therefore, by the spectral theory of compact operators, its spectrum σ(V) = .^{[2] [3]}
• V is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero), but it is not nilpotent operator.

See also

• Product integral

Further reading

• Book: Gohberg, Israel . M. G. . Krein . Theory and Applications of Volterra Operators in Hilbert Space . Providence . American Mathematical Society . 1970 . 0-8218-3627-7 .

Notes and References

1. Book: Rynne. Bryan P.. Youngson. Martin A.. Linear Functional Analysis. Springer. 2008. Integral and Differential Equations 8.2. Volterra Integral Equations. 245.
2. Web site: May 30, 2012 . Spectrum of Indefinite Integral Operators . Stack Exchange.
3. Web site: Volterra Operator is compact but has no eigenvalue . .