Problem 35 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

If \(A\) is a symmetric operator, show that \(A^{*}\) is symmetric if and only if it is self-edjoint, \(A^{*}=A^{* *}\)

Step-by-Step Solution

Answer

Given a symmetric operator \(A\), its adjoint can be said to be symmetric if and only if it's self-adjoint. This is proved by taking arbitrary vectors \(x, y\), applying the properties of symmetric and self-adjoint operators, and displaying that both resultant expressions can be equated due to the interchangeability of the inner product.

Step 1 - Define Given and Terms

A linear operator \(A\) on an inner product space is defined as symmetric if \(\langle Ax, y\rangle = \langle x, Ay\rangle\) for every vector \(x, y\). The adjoint of an operator \(A\), denoted as \(A^{*}\), is a unique operator such that \(\langle Ax, y\rangle = \langle x, A^{*}y\rangle\) for every vector \(x, y\). \(A^{*}\) is self-adjoint if and only if \(A^{*}=A^{* *}\).

Step 2 - Prove Direction

Start by showing the direction that if \(A^{*}\) is symmetric, then it is self-adjoint. Here, let's take any vectors \(x, y\), then we can say \(\langle A^{*}x, y\rangle = \langle x, A^{*}y \rangle\). According to the definition of the adjoint operator, it can be rewritten as \(\langle x, A^{**}y \rangle\). Hence, proving that \(A^{*}\) is self-adjoint i.e., \(A^{*}=A^{**}\).

Step 3 - Prove the Reverse Direction

Now, assume that the adjoint operator \(A^{*}\) is self-adjoint. Therefore, \(A^{*} = A^{* *}\). Then, \(\langle x, A^{*}y \rangle = \langle x, A^{**}y \rangle\) which according to the definition of the adjoint operator can be rewritten as \(\langle A^{*}x, y \rangle\). Hence proving that if \(A^{*}\) is self-adjoint, then it is symmetric.

Key Concepts

Self-Adjoint Operator

In linear algebra, understanding the concept of a self-adjoint operator is crucial for analyzing operators on inner product spaces. A self-adjoint operator is essentially an operator that is its own adjoint. If we have an operator denoted as \( A \), its adjoint, represented as \( A^* \), satisfies the relationship \( A = A^* \).
This means that applying such an operator impacts a vector in the same way as its adjoint would.
To determine if an operator \( A^* \) is self-adjoint, one must check that its double adjoint is equal to itself, i.e., \( A^* = A^{**} \).
This property ties deeply into the workings of symmetric operators where the inner product specifications are met in both directions. Understanding this aspect allows students to bridge the concepts of symmetry and self-adjoint operators.
For example, if a symmetric operator \( A \) is given as symmetric, then its adjoint \( A^* \) is self-adjoint if and only if \( A^* = A^{**} \).
This provides a fascinating path from the properties of symmetric operators to self-adjoint operators, enriching your understanding of their significance in inner product spaces.

Adjoint Operator

The adjoint operator is a powerful concept in linear algebra, used to help determine important properties of operators in inner product spaces.
When you have a linear operator \( A \), its adjoint \( A^* \) is defined so that it behaves according to the inner product formula.

• The main characteristic of an adjoint operator \( A^* \) is that for any two vectors \( x \) and \( y \), the following relation holds: \( \langle Ax, y \rangle = \langle x, A^*y \rangle \).
• Uniqueness is a key feature; each operator has a unique adjoint.

This relation ensures an operator's consistency with its adjoint, meaning any transformation under \( A \) must also comply with the transformation under its adjoint \( A^* \) in an inner product space.
In practice, the adjoint operator plays a role in various applications such as quantum mechanics and functional analysis, where it assists in understanding complex transformations. Through exploring adjoint operators, students gain insight into the underlying symmetries of linear transformations, helping them connect theory with practical applications.

Inner Product Space

An inner product space is foundational for discussing concepts like symmetric and adjoint operators.
Understanding this type of space helps in visualizing the terms and conditions of operations.

• An inner product space is a vector space equipped with an additional structure called an inner product.
• The inner product is a function that allows you to compute angles and lengths, resembling a dot product in Euclidean spaces.

The inner product is typically written as \( \langle x, y \rangle \), where \( x \) and \( y \) are vectors within that space.
This notation signifies how two vectors interact or how they correlate.
In these spaces, examining operators and their properties is enhanced, allowing students to analyze not only the magnitude but also the directionality of vectors affected by operators like the adjoint or symmetric ones.
Mastery of inner product spaces is essential for advanced mathematical studies, providing a framework for exploring and proving key concepts related to self-adjointness and adherence to symmetry.