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The positive-definite integral operator has kernel formproven 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 with .
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Rynne, Bryan P.; Youngson, Martin A. (2008). "Integral and Differential Equations 8.2. Volterra Integral Equations". Linear Functional Analysis. Springer. p. 245.
Gohberg, Israel; Krein, M. G. (1970). Theory and Applications of Volterra Operators in Hilbert Space. Providence: American Mathematical Society. ISBN0-8218-3627-7.