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A bound for the remainder of the Hilbert-Schmidt series and other results on representation of solutions to the functional equation of the second kind with a self-adjoint compact operator as an infinite series. (English) Zbl 0830.47007

Summary: For the functional equation of the second kind \[ \phi- \lambda{\mathbf K} \phi= {\mathbf f},\tag{1} \] with \({\mathbf K}\) a compact self-adjoint linear operator on a Hilbert space (a Fredholm integral equation of the second kind, for example), a bound for the remainder of the Hilbert-Schmidt series is found. It is shown that the series solution to (1) introduced in the author’s previous paper [Appl. Math. Lett. 7, No. 6, 71-74 (1994; Zbl 0819.47030)] is (much) more rapidly convergent than the Hilbert- Schmidt series and generally speaking, is a preferable way of expressing the solution to (1) for regular \(\lambda\) as an infinite series. Other series solutions to (1) are given. The corresponding expressions for the inverse \(({\mathbf I}- \lambda {\mathbf K})^{- 1}\) and the resolvent \({\mathbf B}_\lambda\), and also for the resolvent of the Fredholm integral equation of the second kind with symmetric kernel, are given too.

MSC:

47A50 Equations and inequalities involving linear operators, with vector unknowns
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)

Citations:

Zbl 0819.47030
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References:

[1] Tselnik, D. S., A simple bound for the remainder of the Neumann series in the case of a self-adjoint compact operator, Appl. Math. Lett., 7, 6, 71-74 (1994) · Zbl 0819.47030
[2] Kantorovich, L. V.; Akilov, G. P., Functional Analysis (1982), Pergamon Press: Pergamon Press Oxford · Zbl 0484.46003
[3] Mikhlin, S. G., Linear Integral Equations (1960), Hindustan Publishing Corporation: Hindustan Publishing Corporation New Delhi · Zbl 0142.39201
[4] Tselnik, D. S., Problem of a jet flowing into the surface of a heavy liquid, Fluid Dynamics, 8, 2, 244-250 (1973)
[5] Tselnik, D. S., Constructing the Neumann series—An example, Elem. Math., 46, 6, 165-169 (1991) · Zbl 0752.45001
[6] Tselnik, D. S., A series solution for Fredholm integral equation of the second kind with symmetric kernel, Abstracts Amer. Math. Soc., 15, 1, 92 (1994)
[7] Tselnik, D. S., One more series solution for Fredholm integral equation with symmetric kernel, Abstracts Amer. Math. Soc., 15, 1, 237 (1994)
[8] Tselnik, D. S., Series solution for Fredholm integral equation of the second kind with symmetric kernel valid for regular values of λ on the disk |\(λ\)| < \(λ_{m + 1}\)|, Abstracts Amer. Math. Soc., 15, 2, 337 (1994)
[9] Tselnik, D. S., Series solutions and estimate for remainder of the Neumann series for functional equation of the second kind with self-adjoint operator, Abstracts Amer. Math. Soc., 15, 3, 403 (1994) · Zbl 0819.47030
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