Direct approach for the characteristic function of a dissipative operator with distributional potentials
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Direct approach for the characteristic function of a dissipative operator with distributional potentials 1 Ekin Ugurlu ˘
Received: 22 July 2018 / Revised: 22 July 2018 / Accepted: 1 August 2020 © Springer Nature Switzerland AG 2020
Abstract The main aim of this paper is to investigate the spectral properties of a singular dissipative differential operator with the help of its Cayley transform. It is shown that the Cayley transform of the dissipative differential operator is a completely non-unitary contraction with finite defect indices belonging to the class C0 . Using its characteristic function and the spectral properties of the resolvent operator, the complete spectral analysis of the dissipative differential operator is obtained. Embedding the Cayley transform to its natural unitary colligation, a Carathéodory function is obtained. Moreover, the truncated CMV matrix is established which is unitary equivalent to the Cayley transform of the dissipative differential operator. Furthermore, it is proved that the imaginary part of the inverse operator of the dissipative differential operator is a rank-one operator and the model operator of the associated dissipative integral operator is constructed as a semi-infinite triangular matrix. Using the characteristic function of the dissipative integral operator with rank-one imaginary component, associated Weyl functions are established. Keywords Completely non-unitary contraction · Dissipative operator · Characteristic function · Spectral analysis · CVM matrix · Jacobi operator Mathematics Subject Classification Primary 47A45; Secondary 47B44 · 34B05 · 34B37 · 34B40 · 47B36
1 Introduction The spectral properties of the completely non-unitary contraction (c.n.u.) belonging to the class C0 has been studied by Sz.-Nagy and Foia¸s [1] (see also [2]). One of
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Ekin U˘gurlu [email protected] Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, 06530 Balgat, Ankara, Turkey 0123456789().: V,-vol
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the main tools in such investigation is the minimal function m T of the contraction T . The minimal function m T provides some information about the spectrum of the contraction T belonging to the class C0 that consists of those c.n.u. contractions T for which there exists a non-zero function u ∈ H ∞ (H p denotes the Hardy class) such that u(T ) = 0. It is known that such a function u has a canonical factorization into the product of an inner function u i and outer function u e , u(T ) = 0 implies u i (T ) = 0. Sz.-Nagy and Foia¸s proved that for every contraction T ∈ C0 there exists a minimal function m T which is the inner function satisfying m T (T ) = 0 such that every other functions v ∈ H ∞ with v(T ) = 0 is a multiple of m T . This minimal function m T is determined up to a constant factor of modulus one. The zeros of minimal function m T in the open disc D and of the complement, in the unit circle C, of the union of the arcs of C on which m T is analytic, and the spectrum of the contraction T ∈ C0 coincide wi
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