We study the relationship between the adjacency matrix and the discrete Laplacian acting on 1-forms. We also prove that if the adjacency matrix is bounded from below it is not necessarily essentially self-adjoint. We discuss the question of essential self-adjointness and the notion of completeness.2010 Mathematics Subject Classification. 81Q35, 47B25, 05C63.
Muraleetharan and Thirulogasanthan in (J. Math. phys. 59, No. 10, 103506, 27p. (2018)) introduced the concept of Calkin spherical spectrum of a bounded quaternionic linear operators. The study of this spectrum is establisched using the Fredholm operators theory. Motivated by this, we study the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism. First, we investigate the Fredholm spherical spectrum of the sum of two elements in quaternionic Banach algebra by means of the Fredholm spherical spectrum of the two elements. Next, we prove a perturbation result on this spectrum. We also study the boundary spherical spectrum. As application, we investigate the Fredholm and Weyl spherical spectra of bounded right quaternionic linear operators.
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