An impressive method for adjoint of linear and nonlinear operators

Document Type : Original Article

Authors

1 Department of Physics, Amirkabir University of Technology (Tehran Polytechnic ), P.O.Box: 15875-4413,Tehran, Iran

2 Department of Physics, Faculty of Physics & Chemistry, Alzahra University, Tehran, Iran

Abstract

Abstract In this paper we have obtained the adjoint of an arbitrary operator (linear and nonlinear) in Hilbert space by introducing n-dimensional Riemannian manifold. This general formalism covers every linear operator (non – dif-ferential) in Hilbert space. In fact, our approach shows that instead of using directly the adjoint definition of an operator, it can be obtained directly by relying on a suitable generalized space according to the action of the operator in question. For the case of nonlinear operators we have to change the definition of the linear operator adjoint. But here, we have obtained adjoint of these operators with respect to the definition of the derivative of the operator. In matter fact, we have shown one of the straight applications of ''Frechet derivative'' into algebra of the operators.
1 Introduction
This paper consists of two main parts. In first part, we look for a general relationship for the adjoint of the linear operators. In fact, we intend to achieve a universal formula for the adjoint of the linear operators with a generalized space such as Riemannian manifold. Although the self-adjoint extensions of differential operators on Riemannian manifold has been studied [1] but our main focus is on non - differential operators. Recently, it has been reported for linear operators to be unitary in curved space [2]
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Volume 1, Issue 1
February 2024
Pages 28-32
  • Receive Date: 08 July 2023
  • Revise Date: 16 December 2023
  • Accept Date: 02 January 2024
  • First Publish Date: 02 January 2024
  • Publish Date: 01 February 2024