%0 Journal Article
%T An Alternate Proof of the De Branges Theorem on Canonical Systems
%A Keshav Raj Acharya
%J ISRN Mathematical Analysis
%D 2014
%R 10.1155/2014/704607
%X The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on . This provides an alternative proof of the De Branges theorem that the canonical systems with imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system. 1. Introduction This paper deals with the canonical systems of the following form: Here and is a positive semidefinite matrix whose entries are locally integrable. For fixed , a function is called a solution if is absolutely continuous and satisfies (1). Consider the Hilbert space as follows: provided with an inner product . The canonical systems (1) on have been studied by Hassi et al., Winkler, and Remling in [1¨C4] in various contexts. The Jacobi and Schr£żdinger equations can be written into canonical systems with appropriate choice of . In addition, canonical systems are closely connected with the theory of the De Branges spaces and the inverse spectral theory of one-dimensional Schr£żdinger operators; see [3]. We believe that the extensions of the theories from these equations to the canonical systems are to be of general interest. If the system (1) can be written in the form of then we may consider this as an eigenvalue equation of an operator on . But is not invertible in general. Instead, the system (1) induces a linear relation that may have a multivalued part. Therefore, we consider this as an eigenvalue problem of a linear relation induced by (1) on . For some , if the canonical system (1) has all solutions in , we say that the system is in the limit circle case, and if the system has unique solution in , we say that the system is in limit point case. The basic results in this paper are the following theorems. Theorem 1. In the limit circle case, the defect index of the symmetric relation , induced by (1), is constant on . The immediate consequence of the Theorem 1 is the following theorem. Theorem 2 (De Branges). The canonical systems with prevail the limit point case. Theorem 2 has been proved in [5] by function theoretic approach. However the proof was not easily readable to me and we thought of providing an alternate and simple proof of the theorem. In order to prove the main theorems we use the results from the papers [1, 3, 4] and we use the spectral theory of a linear relation from [6]. Let be a Hilbert space over and denote by the Hilbert space . A linear relation on is a subspace of . The adjoint of on is a closed linear relation defined by A linear relation is called
%U http://www.hindawi.com/journals/isrn.mathematical.analysis/2014/704607/