Partially fundamentally reducible operators in Krein spaces

Abstract

A self-adjoint operator A in a Kreȋn space (K, [ • , • ]) is called partially fundamentally reducible if there exist a fundamental decomposition K = K + [ +]K − (which does not reduce A) and densely defined symmetric operators S + and S − in the Hilbert spaces (K + , [ • , • ]) and (K − , −[ • , • ]), respectively, such that each S + and S − has defect numbers (1, 1) and the operator A is a self-adjoint extension of S = S + ⊕ (−S − ) in the Kreȋn space (K, [ • , • ]). The operator A is interpreted as a coupling of operators S + and −S − relative to some boundary triples C, Γ + 0 , Γ + 1 and C, Γ − 0 , Γ − 1 . Sufficient conditions for a nonnegative partially fundamentally reducible operator A to be similar to a self-adjoint operator in a Hilbert space are given in terms of the Weyl functions m + and m − of S + and S − relative to the boundary triples C, Γ + 0 , Γ + 1 and C, Γ − 0 , Γ − 1 . Moreover, it is shown that under some asymptotic assumptions on m + and m − all positive self-adjoint extensions of the operator S are similar to self-adjoint operators in a Hilbert space.