Hankel operators in the perturbation theory of unbounded self-adjoint operators. (English) Zbl 0716.47015
Analysis and partial differential equations, Coll. Pap. dedic. Mischa Cotlar, Lect. Notes Pure Appl. Math. 122, 529-544 (1990).
[For the entire collection see Zbl 0684.00014.]
The author generalizes his results on perturbation theory of bounded selfadjoint operators to unbounded selfadjoint operators, the proofs are new and very laborious.
Main result: Let f be a function of the Besov class \(B^ 1_{\infty}({\mathbb{R}})\). If A, B are unbounded selfadjoint operators on Hilbert space such that A-B is bounded, then: \[ \| f(A)-f(B)\| \leq C\| f\|_{B^ 1_{\infty 1}({\mathbb{R}})}\| A-B\|. \] In addition, if A-B is of trace class, then f(A)-f(B) is of trace class, and the following trace formula holds: \[ trace(f(A)- f(B))=\int^{\infty}_{-\infty}s(t)f'(t)dt, \] where s is the spectral shift of the pair (A,B).
The author generalizes his results on perturbation theory of bounded selfadjoint operators to unbounded selfadjoint operators, the proofs are new and very laborious.
Main result: Let f be a function of the Besov class \(B^ 1_{\infty}({\mathbb{R}})\). If A, B are unbounded selfadjoint operators on Hilbert space such that A-B is bounded, then: \[ \| f(A)-f(B)\| \leq C\| f\|_{B^ 1_{\infty 1}({\mathbb{R}})}\| A-B\|. \] In addition, if A-B is of trace class, then f(A)-f(B) is of trace class, and the following trace formula holds: \[ trace(f(A)- f(B))=\int^{\infty}_{-\infty}s(t)f'(t)dt, \] where s is the spectral shift of the pair (A,B).
Reviewer: B.D.Khanh
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 47A55 | Perturbation theory of linear operators |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
| 47A60 | Functional calculus for linear operators |
| 47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |
