Cloning of Symmetric Operators
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Complex Analysis and Operator Theory
Cloning of Symmetric Operators Yu. M. Arlinski˘ı1 Received: 3 July 2020 / Accepted: 22 October 2020 © Springer Nature Switzerland AG 2020
Abstract Given a closed densely defined symmetric operator S in a separable Hilbert space, by means of a non-real and non-imaginary complex number z and the corresponding deficiency subspace Nz of S we define a new closed symmetric operator S(z) with dense domain. We prove that the operator S(z) preserves various properties of S. When the deficiency indices of S are equal a bijection of the set of all selfadjoint extensions of S onto the set of all selfadjoint extensions of S(z) is established. We consider in detail the case when a symmetric operator S is nonnegative. Keywords Symmetric operator · Deficiency subspace · Selfadjoint operator · Clone Mathematics Subject Classification Primary 47B25; Secondary 47A20
1 Introduction Notations. We use the symbols dom T , ran T , ker T for the domain, the range, and the null-subspace of a linear operator T . The closures of dom T , ran T are denoted by dom T , ran T , respectively. The identity operator in a Hilbert space H is denoted by I and sometimes by IH . If L is a subspace, i.e., a closed linear manifold in H, the orthogonal projection in H onto L is denoted by PL . The notation T L means the restriction of a linear operator T on the set L ⊂ dom T . The resolvent set of T is
Communicated by Seppo Hassi. Dedicated to Henk de Snoo on the occasion of his 75th birthday. This article is part of the topical collection “Recent Developments in Operator Theory - Contributions in Honor of H.S.V. de Snoo” edited by Jussi Behrndt and Seppo Hassi.
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Yu. M. Arlinski˘ı [email protected] Stuttgart, Germany 0123456789().: V,-vol
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denoted by ρ(T ). C and R denote the fields of complex and real numbers, respectively, R+ := [0, +∞), N is the set of all natural numbers. Let S be a closed densely defined symmetric operator in the complex infinitedimensional Hilbert space H. Set Mλ := ran (S − λI ) = (S − λI )dom S, Nλ := H Mλ = ker(S ∗ − λI ), λ ∈ C. The subspace Nλ is called the deficiency subspace of S [1] and the numbers n + = dim ker(S ∗ − λI ), Im λ > 0, n − = dim ker(S ∗ − λI ), Im λ < 0. are called the deficiency indices (numbers). We denote by PMλ and PNλ orthogonal projections onto Mλ and Nλ , respectively. Clearly, Mλ¯ ⊕ Nλ = H ⇐⇒ PMλ¯ + PNλ = IH . According to J. von Neumann results [1] (1) The domain of the adjoint operator S ∗ admits the following direct decomposition ˙ λ +N ˙ λ¯ , Im λ = 0. dom S ∗ = dom S +N (2) S admits selfadjoint extensions in H if and only if the deficiency indices of S are equal. A symmetric operator S is called simple (or prime) if there is no reducing subspace on which S is selfadjoint. It is well known that S is simple ⇐⇒ span {Nλ , λ ∈ C \ R} = H ⇐⇒
Mλ = {0}.
λ∈C\R
Definition 1.1 Let S be a closed densely defined symmetric operator and let z be a complex number such that z ∈ C \ {R ∪ iR}, Nz = {0}. Then define the
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