The Operator Norm of Perturbation of the Identity

Oct 13, 2025 by talynt.org 139 views

Introduction

In the realm of functional analysis, operator theory, and normed spaces, the concept of the operator norm of perturbation of the identity plays a crucial role in understanding the behavior of linear operators on Hilbert spaces. This article delves into the discussion of the operator norm of perturbation of the identity, exploring its properties and implications.

Preliminaries

Let $\mathscr{H}$ be a separable Hilbert space, and let $T \in \mathscr{B}(\mathscr{H})$ be a bounded linear operator on $\mathscr{H}$ . The operator norm of $T$ is defined as $\|T\| = \sup\{\|Tx\| : x \in \mathscr{H}, \|x\| \leq 1\}$ . The identity operator on $\mathscr{H}$ is denoted by $\mathrm{Id}$ , and its norm is $\|\mathrm{Id}\| = 1$ .

The Operator Norm of Perturbation of the Identity

The operator norm of perturbation of the identity is defined as $\|\mathrm{Id} + T\| = \sup\{\|(\mathrm{Id} + T)x\| : x \in \mathscr{H}, \|x\| \leq 1\}$ . The given condition $\|\mathrm{Id} + T\| = 1 + \|T\|$ is a crucial assumption in this discussion.

Necessity of the Condition

To prove that $T$ satisfies the condition $\|\mathrm{Id} + T\| = 1 + \|T\|$ if and only if the operator $\mathrm{Id} + T$ is invertible, we start by assuming that $\|\mathrm{Id} + T\| = 1 + \|T\|$ . We need to show that $\mathrm{Id} + T$ is invertible.

Step 1: Showing that $\mathrm{Id} + T$ is one-to-one

Suppose that $(\mathrm{Id} + T)x = 0$ for some $x \in \mathscr{H}$ . Then, we have $x + Tx = 0$ , which implies that $x = -Tx$ . Since $\|x\| \leq 1$ , we have $\|-Tx\| \leq 1$ , which implies that $\|Tx\| \leq 1$ . Therefore, $\|T\| \leq 1$ , and hence $\|\mathrm{Id} + T\| \leq 2$ . This contradicts the assumption that $\|\mathrm{Id} + T\| = 1 + \|T\|$ . Therefore, $\mathrm{Id} + T$ is one-to-one.

Step 2: Showing that $\mathrm{Id} + T$ is onto

Since $\mathscr{H}$ is a Hilbert space, it is complete. Let $y \in \mathscr{H}$ be arbitrary. We need to show that there exists $x \in \mathscr{H}$ such that $(\mathrm{Id} + T)x = y$ . Since $\mathrm{Id} + T$ is one-to-one, we can define a linear functional $f_y$ on the range of $\mathrm{Id} + T$ by $f_y((\mathrm{Id} + T)x) = \langle y, x \rangle$ . We need to show that $f_y$ is bounded.

Step 3: Showing that $f_y$ is bounded

Since $\mathrm{Id} + T$ is one-to-one, we have $\|(\mathrm{Id} + T)x\| \geq \|x\|$ for all $x \in \mathscr{H}$ . Therefore, $\|f_y((\mathrm{Id} + T)x)\| = |\langle y, x \rangle| \leq \|y\| \cdot \|x\| \leq \|y\| \cdot \|(\mathrm{Id} + T)x\|$ . This implies that $f_y$ is bounded, and hence $f_y$ is a bounded linear functional on the range of $\mathrm{Id} + T$ .

Step 4: Showing that $\mathrm{Id} + T$ is invertible

Since $\mathscr{H}$ is a Hilbert space, the range of $\mathrm{Id} + T$ is closed. Therefore, the bounded linear functional $f_y$ on the range of $\mathrm{Id} + T$ has a bounded extension to the whole space $\mathscr{H}$ . This implies that there exists $x \in \mathscr{H}$ such that $f_y((\mathrm{Id} + T)x) = \langle y, x \rangle$ . Therefore, $(\mathrm{Id} + T)x = y$ , and hence $\mathrm{Id} + T$ is onto.

Sufficiency of the Condition

Conversely, suppose that $\mathrm{Id} + T$ is invertible. We need to show that $\|\mathrm{Id} + T\| = 1 + \|T\|$ . Since $\mathrm{Id} + T$ is invertible, we have $\|\mathrm{Id} + T\| \geq 1$ . Therefore, $\|\mathrm{Id} + T\| \geq 1 + \|T\|$ .

Step 1: Showing that $\|\mathrm{Id} + T\| \leq 1 + \|T\|$

Suppose that $\|x\| \leq 1$ . Then, we have $\|(\mathrm{Id} + T)x\| = \|(x + Tx)\| \leq \|x\| + \|Tx\| \leq 1 + \|T\|$ . Therefore, $\|\mathrm{Id} + T\| \leq 1 + \|T\|$ .

Conclusion

In conclusion, we have shown that the operator norm of perturbation of the identity is equal to $1 + \|T\|$ if and only if the operator $\mathrm{Id} + T$ is invertible. This result has important implications in the study of linear operators on Hilbert spaces.

References

[1] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, 1983.
[2] E. Nelson, Notes on Non-Relativistic Quantum Mechanics, Princeton University Press, 1964.
[3] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, 1972.

Introduction

In our previous article, we explored the concept of the operator norm of perturbation of the identity and its properties. In this article, we will address some frequently asked questions related to this topic.

Q: What is the significance of the operator norm of perturbation of the identity?

A: The operator norm of perturbation of the identity is a fundamental concept in operator theory, and it plays a crucial role in understanding the behavior of linear operators on Hilbert spaces. It has important implications in the study of linear operators, and it is used in various areas of mathematics, such as functional analysis, operator theory, and normed spaces.

Q: What is the relationship between the operator norm of perturbation of the identity and the invertibility of the operator?

A: We have shown that the operator norm of perturbation of the identity is equal to $1 + \|T\|$ if and only if the operator $\mathrm{Id} + T$ is invertible. This result has important implications in the study of linear operators on Hilbert spaces.

Q: How does the operator norm of perturbation of the identity relate to the norm of the operator?

A: We have shown that the operator norm of perturbation of the identity is equal to $1 + \|T\|$ . This means that the norm of the operator $\mathrm{Id} + T$ is equal to the sum of the norms of the identity operator and the operator $T$ .

Q: What are some applications of the operator norm of perturbation of the identity?

A: The operator norm of perturbation of the identity has important applications in various areas of mathematics, such as functional analysis, operator theory, and normed spaces. It is used in the study of linear operators, and it has implications in the study of Hilbert spaces.

Q: How can I use the operator norm of perturbation of the identity in my research?

A: The operator norm of perturbation of the identity is a fundamental concept in operator theory, and it can be used in various areas of research. You can use it to study the behavior of linear operators on Hilbert spaces, and it can be used to derive important results in functional analysis, operator theory, and normed spaces.

Q: What are some common mistakes to avoid when working with the operator norm of perturbation of the identity?

A: When working with the operator norm of perturbation of the identity, it is essential to avoid common mistakes, such as:

Assuming that the operator norm of perturbation of the identity is always equal to $1 + \|T\|$ .
Failing to check the invertibility of the operator $\mathrm{Id} + T$ .
Not considering the implications of the operator norm of perturbation of the identity on the norm of the operator.

Conclusion

In conclusion, the operator norm of perturbation of the identity is a fundamental concept in operator theory, and it has important implications in the study of linear operators on Hilbert spaces. We hope that this Q&A article has provided you with a better understanding of this concept and its applications.

References

[1] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, 1983.
[2] E. Nelson, Notes on Non-Relativistic Quantum Mechanics, Princeton University Press, 1964.
[3] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Academic Press, 1972.