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We give a simple definition of a spectral shift function for pairs of nonpositive operators on Banach spaces and prove trace formulas of Lifshitz-Kreȋn type for a perturbation of an operator monotonic (negative complete Bernstein) function of negative and nonpositive operators on Banach spaces induced by nuclear perturbation of an operator argument. The Lipschitzness of such functions is also investigated. The results may be regarded as a contribution to a perturbation theory for Hirsch functional calculus.Key wards. Spectral shift function, Lifshitz-Kreȋn trace formula, Hirsch functional calculus, negative operator, Banach space, perturbation determinant where c ≤ 0, b ≥ 0 and µ is a unique positive measure that satisfies the condition (0,∞) dµ(t)/(1+ t) < ∞.A lot of examples of complete Bernstein functions one can found in [33].In the sequel unless otherwise stated we assume for the sake of simplicity that c = b = 0 in the integral representation (1) (otherwise one should replace ϕ(z) by ϕ(z) − c − bz).Remark 2.2 It is known (see, e.g., [33, Theorem 12.17]), that the families of complete Bernstein and positive operator monotone functions coincide. It follows that the families of negative complete Bernstein and negative operator monotone functions also coincide (we say that a real Borel function ϕ on (−∞, 0] is negative operator monotone if for every bounded self-adjoint operators A and B on a finite or infinite-dimensional real Hilbert space the inequalities ϕ(B)). That is why we denote the family of negative complete Bernstein functions by OM − . Definition 2.3 We say that (closed, densely defined) operator A on a complex Banach space X is nonpositive (negative) if (0, ∞) is contained in ρ(A), the resolvent set of A, andwhere R(t, A) = (tI − A) −1 stands for the resolvent of an operator A, and Ix = x for all x ∈ X.So, the operator A is nonpositive (negative) if and only if −A is non-negative (respectively positive) in a sense of Komatsu [14] (see also [19, Chapter 1]). We denote by N P(X) (respectively N (X)) the class of nonpositive (negative) operators on X. (We deal with negative operators instead of positive one because in this form our results are consistent with the multidimensional Bochner-Phillips functional calculus of semigroup generators built in [21] -[26].) Note that A − εI ∈ N (X) for A ∈ N P(X), ε > 0.Since every non-negative (closed, densely defined) operator on X is sectorial of angle ω for some ω ∈ (0, π) (see, e.g., [19, Proposotion 1.2.1]) every operator A ∈ N P(X) enjoys the following properties (S ω denotes the open sector symmetric about the positive real axis with opening angle 2ω):(i) the resolvent set ρ(A) contains the sector S ω , ω = arcsin 1/M A ; Consequently, ifA is negative then (i ′ ) the resolvent set ρ(A) contains the closure of some set of the form S θ ∪ B δ (0) (0 < θ < arcsin 1/M A ; B δ (0) stands for the open disc centered at zero of radius δ > 0); (ii ′ ) there is some constant M ′ A ≥ M A such that R(λ, A) ≤ M ′ A 1 + |λ| for λ in some neighborhood of the closure of S...
We give a simple definition of a spectral shift function for pairs of nonpositive operators on Banach spaces and prove trace formulas of Lifshitz-Kreȋn type for a perturbation of an operator monotonic (negative complete Bernstein) function of negative and nonpositive operators on Banach spaces induced by nuclear perturbation of an operator argument. The Lipschitzness of such functions is also investigated. The results may be regarded as a contribution to a perturbation theory for Hirsch functional calculus.Key wards. Spectral shift function, Lifshitz-Kreȋn trace formula, Hirsch functional calculus, negative operator, Banach space, perturbation determinant where c ≤ 0, b ≥ 0 and µ is a unique positive measure that satisfies the condition (0,∞) dµ(t)/(1+ t) < ∞.A lot of examples of complete Bernstein functions one can found in [33].In the sequel unless otherwise stated we assume for the sake of simplicity that c = b = 0 in the integral representation (1) (otherwise one should replace ϕ(z) by ϕ(z) − c − bz).Remark 2.2 It is known (see, e.g., [33, Theorem 12.17]), that the families of complete Bernstein and positive operator monotone functions coincide. It follows that the families of negative complete Bernstein and negative operator monotone functions also coincide (we say that a real Borel function ϕ on (−∞, 0] is negative operator monotone if for every bounded self-adjoint operators A and B on a finite or infinite-dimensional real Hilbert space the inequalities ϕ(B)). That is why we denote the family of negative complete Bernstein functions by OM − . Definition 2.3 We say that (closed, densely defined) operator A on a complex Banach space X is nonpositive (negative) if (0, ∞) is contained in ρ(A), the resolvent set of A, andwhere R(t, A) = (tI − A) −1 stands for the resolvent of an operator A, and Ix = x for all x ∈ X.So, the operator A is nonpositive (negative) if and only if −A is non-negative (respectively positive) in a sense of Komatsu [14] (see also [19, Chapter 1]). We denote by N P(X) (respectively N (X)) the class of nonpositive (negative) operators on X. (We deal with negative operators instead of positive one because in this form our results are consistent with the multidimensional Bochner-Phillips functional calculus of semigroup generators built in [21] -[26].) Note that A − εI ∈ N (X) for A ∈ N P(X), ε > 0.Since every non-negative (closed, densely defined) operator on X is sectorial of angle ω for some ω ∈ (0, π) (see, e.g., [19, Proposotion 1.2.1]) every operator A ∈ N P(X) enjoys the following properties (S ω denotes the open sector symmetric about the positive real axis with opening angle 2ω):(i) the resolvent set ρ(A) contains the sector S ω , ω = arcsin 1/M A ; Consequently, ifA is negative then (i ′ ) the resolvent set ρ(A) contains the closure of some set of the form S θ ∪ B δ (0) (0 < θ < arcsin 1/M A ; B δ (0) stands for the open disc centered at zero of radius δ > 0); (ii ′ ) there is some constant M ′ A ≥ M A such that R(λ, A) ≤ M ′ A 1 + |λ| for λ in some neighborhood of the closure of S...
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