On compressions of self-adjoint extensions of a symmetric linear relation

2018/12/01 by Mogilevskii, Vadim
#47A06 #47A20 #47A56 #47B25 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1812.00204

Abstract

Let A be a symmetric linear relation in the Hilbert space \gH with equal deficiency indices n_± (A)≤∞. A self-adjoint linear relation \wt A⊃ A in some Hilbert space \wt\gH⊃ \gH is called an exit space extension of A; such an extension is called finite-codimensional if dim (\wt\gH\ominus\gH)< ∞. We study the compressions C (\wt A)=P_\gH\wt A\up\gH of exit space extensions \wt A=\wt A^*. For a certain class of extensions \wt A we parameterize the compressions C (\wt A) by means of abstract boundary conditions. This enables us to characterize various properties of C (\wt A) (in particular, self-adjointness) in terms of the parameter for \wt A in the Krein formula for resolvents. We describe also the compressions of a certain class of finite-codimensional extensions. These results develop the results by A. Dijksma and H. Langer obtained for a densely defined symmetric operator A with finite deficiency indices.

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