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 huge device for fixing linear equations and invertibility stipulations for linear operators, permit us to explain the spectra, to guage the multiplicities of eigenvalues, and so forth. We derive top and reduce bounds, and perturbation effects for determinants, and talk 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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2), |det (I + A + B)|2 ≤ det (I + MA + MB )det (I + MA∗ + MB ∗ ), since n n (1 + sk (A)) = k=1 (1 + sk (A∗ )). k=1 So it suffices to consider the case operators A, B selfadjoint and positive definite. But then det (I + A + B) = det ((I + A)1/2 (I + (I + A)−1/2 B(I + A)−1/2 )(I + A)1/2 ) = = det (I + A) det (I + (I + A)−1/2 B(I + A)−1/2 ). Take into account that Therefore (I + A)−1/2 ≤ I and (I + A)1/2 ≥ I. (I + A)−1/2 B(I + A)−1/2 ≤ B. Thus det (I + (I + A)−1/2 B(I + A)−1/2 ) ≤ det (I + B). This proves the theorem.

194), the inequality |detp (I − A)| ≤ exp qp Npp (A) (p ≥ 1) is presented but the constant qp for p > 2 is unknown. 2 refines the latter inequality. 7 is adopted from the papers (Gil’ 2008) and (Gil’ 2013). About additional lower bounds for determinants of the trace class operators see the paper (Bandtlow and G¨ uven 2015), in which the authors have derived explicit upper bounds for the spectral distance of two trace class operators. 6 Nakano type operators . . . . . . . . . . . . .

Q. E. D. 2. ), ln k then A is of infinite order. Take pk = k: γπ (A) := ∞ j=1 Since ∞ k=2 sjj (A) . j tk < ∞ (t > 0), k(ln k)k we have γπ (tA) < ∞. 2 Bounds for eigenvalues of Nakano operators Put τA = 1 A H if A if A H H ≤ 1, > 1. ) of A are enumerated with the multiplicities in the decreasing way of their absolute values. 1 Let A be compact. ). In particular, if γπ (A/τA ) < ∞, then ∞ j=1 |λj (A)|pj ≤ γπ (A/τA ). 2). , xn ) (0 < xn ≤ xn−1 ≤ ... , n), ∂xk 46 3. , n − 1). , sn ), where sk = sk (A), λk = λk (A).