March 7, 2017

Bounds for determinants of linear operators and their by Michael Gil'

By Michael Gil'

This e-book bargains with the determinants of linear operators in Euclidean, Hilbert and Banach areas. Determinants of operators supply us a tremendous software for fixing linear equations and invertibility stipulations for linear operators, allow us to explain the spectra, to guage the multiplicities of eigenvalues, and so forth. We derive top and decrease bounds, and perturbation effects for determinants, and speak about purposes of our theoretical effects to spectrum perturbations, matrix equations, parameter eigenvalue difficulties, in addition to to differential, distinction and functional-differential equations.

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M m=1 Denote 1 1 ∆p := Np (V− ) exp (1 + Np (A + A+ ) + Np (V− ))p . 1 For an integer p ≥ 1, let A ∈ SNp , and ∆p < |detp (I − D)|. Then I − A is invertible and |detp (I − A)| ≥ |detp (I − D)| − ∆p . 2. ;s∈[0,1] |1 − sajj | > 0. 2, we can write. |detp (I − D)| ≥ exp − Npp (D) . pφD Now the previous theorem implies. 1) and ∆p < exp − Npp (D) pφD hold. 9 Npp (D) − ∆p . 2 are well known. 2). In the well known books (Dunford 1963, p. 1106), (Gohberg, Goldberg and Krupnik 2000, p. 194), the inequality |detp (I − A)| ≤ exp qp Npp (A) (p ≥ 1) is presented but the constant qp for p > 2 is unknown.

3. The characteristic determinant of a nuclear operator 27 which corresponds to the complete orthonormal system exp (2πki). It follows from here that sj (A) = |Cj | and hence ∞ spk (A) = ∞ k=1 for any p < 2 and in particular for p = 1. We complete this section with the following well known theorem (see, for example, (Dunford and Schwarz 1966), (Gohberg and Krein 1969)). 7 Suppose that K(t, s) is a function measurable on [a, b]×[a, b]. 1) is a Hilbert-Schmidt operator in L2 [a, b] if and only if b a b a |K(t, s)|2 dt ds < ∞.

31)). 1 appeared in (Gil’ 2013a). 6 is taken from (Gil’ 2013). 1 is probably new. The following inequality for the determinants of two n × n-matrices A and B is well-known (Bhatia 2007, p. 11) where M2 := max{ A , B }. The spectral norm is unitarily invariant, but often it is not easy to compute that norm. 11). Indeed, for the spectral norm we have αn = 1. If we take A = aB with a positive constant a < 1, then A − B + A + B = 2 B = 2M2 , but 1 n−1 ) ≥ n. n−1 The theory of matrices with dominant principal diagonals is well developed, cf.

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