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Spectral Theorem for a bounded self adjoint operator on a Bicomplex Hilbert space

Akshay Sakharam Rane

Abstract

In this paper, we shall consider the notion of bicomplex inner product and define bicomplex Hilbert space. We shall define $L^{2}[a,b]$ where the functions take bicomplex values. We shall prove the Theorem for a bounded self adjoint operator on a bicomplex Hilbert space which is not compact.

Spectral Theorem for a bounded self adjoint operator on a Bicomplex Hilbert space

Abstract

In this paper, we shall consider the notion of bicomplex inner product and define bicomplex Hilbert space. We shall define where the functions take bicomplex values. We shall prove the Theorem for a bounded self adjoint operator on a bicomplex Hilbert space which is not compact.
Paper Structure (3 sections, 2 theorems, 38 equations)

This paper contains 3 sections, 2 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Bicomplex Inner Product
  3. Main Results

Key Result

Theorem 3.2

Let $H$ be a bi complex Hilbert module. Let $T$ be a linear operator on $H.$ Let $H= e_{1}H_{1}+e_{2}H_{2}$ be its idempotent decomposition. Suppose $H$ has a cyclic vector then $H_{1}=e_1H$ and $H_{2}=eH$ also have a cyclic vector.

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Theorem 3.2
  • proof
  • proof
  • Theorem 3.5
  • proof