On limits of \(L_ p\)-norms of an integral operator. (English) Zbl 0816.47055

Summary: A recurrence relation for the computation of the \(L_ p\)-norms of an Hermitian Fredholm integral operator is derived and an expression giving approximately the number of eigenvalues which in absolute value are equal to the spectral radius is determined. Using the \(L_ p\)-norms for the approximation of the spectral radius of this operator an a priori and an a posteriori bound for the error are obtained. Some properties of the a posteriori bound are discussed.


47G10 Integral operators
47A53 (Semi-) Fredholm operators; index theories
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47A10 Spectrum, resolvent


Full Text: EuDML


[1] L. G. Brown, H. Kosaki: Jensen’s inequality in semi-finite von Neumann algebras. J. Operator Theory 23 (1990), 3-19. · Zbl 0718.46026
[2] A. C. Hearn: REDUCE 2 user’s manual. University of Utah, USA, 1973.
[3] R. A. Kunze: \(L_p\) Fourier transforms on locally compact unimodular groups. Trans. Amer. Math. Soc. 89 (1958), 519-540. · Zbl 0084.33905
[4] C. A. McCarthy: \(c_p\). Israel J. Math. 5 (1967), 249-271.
[5] S. G. Michlin, Ch. L. Smolickij: Approximate methods of solution of differential and integral equations. Nauka, Moscow, 1965.
[6] J. Peetre, G. Sparr: Interpolation and non-commutative integration. Ann. of Math. Pura Appl. 104 (1975), 187-207. · Zbl 0309.46031
[7] I. E. Segal: A non-commutative extension of abstract integration. Ann. of Math. 57 (1953), 401-457, correction 58(1953), 595-596. · Zbl 0051.34202
[8] P. Stavinoha: Convergence of \(L_p\)-norms of a matrix. Aplikace matematiky 30 (1985), 351-360. · Zbl 0609.65024
[9] P. Stavinoha: On limits of \(L_p\)-norms of a linear operator. Czech. Math. J. 32 (1982), 474-480. · Zbl 0511.46062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.