Properties of the Approximate Point Spectrum

Oct 13, 2025 by leolavanderia.com.br 89 views

Introduction

In the realm of functional analysis, the study of bounded operators on Banach spaces is a fundamental area of research. One of the key concepts in this field is the approximate point spectrum of a bounded operator, which plays a crucial role in understanding the spectral properties of such operators. In this article, we will delve into the characterisation of the approximate point spectrum of a bounded operator on a Banach space, exploring its definition, properties, and significance.

Preliminaries

Let $V$ be a complex Banach space, and let $A : V \rightarrow V$ be a bounded operator on $V$ . The approximate point spectrum of $A$ , denoted by $\sigma_{ap}(A)$ , is a set of complex numbers that are "close" to being eigenvalues of $A$ . To be more precise, we say that $k \in \mathbb{C}$ is in the approximate spectrum of $A$ and we write $k \in \sigma_{ap}(A)$ when the following condition holds:

\liminf_{n \rightarrow \infty} \|Ax_n - kx_n\| = 0

for some sequence $\{x_n\}$ in $V$ such that $\|x_n\| = 1$ for all $n$ .

Properties of the Approximate Point Spectrum

The approximate point spectrum of a bounded operator has several interesting properties that make it a useful tool in spectral theory. Some of these properties are listed below:

Non-emptyness: The approximate point spectrum of a bounded operator is always non-empty.
Closedness: The approximate point spectrum of a bounded operator is a closed set in the complex plane.
Invariance under similarity: The approximate point spectrum of a bounded operator is invariant under similarity transformations.
Inclusion in the spectrum: The approximate point spectrum of a bounded operator is included in the spectrum of the operator.

Characterisation of the Approximate Point Spectrum

One of the key questions in the study of the approximate point spectrum of a bounded operator is to characterise the set of complex numbers that belong to this spectrum. In other words, we want to determine the conditions under which a complex number $k$ belongs to the approximate point spectrum of a bounded operator $A$ . The following theorem provides a partial answer to this question:

Theorem 1: Let $A : V \rightarrow V$ be a bounded operator on a complex Banach space $V$ . Then, a complex number $k$ belongs to the approximate point spectrum of $A$ if and only if there exists a sequence $\{x_n\}$ in $V$ such that $\|x_n\| = 1$ for all $n$ , and

\liminf_{n \rightarrow \infty} \|Ax_n - kx_n\| = 0

Proof: The "only if" part of the theorem follows directly from the definition of the approximate point spectrum. To prove the "if" part, we assume that there exists a sequence $\{x_n\}$ in $V$ such that $\|x_n\| = 1$ for all $n$ , and

\liminf_{n \rightarrow \infty} \|Ax_n - kx_n\| = 0

We need to show that $k$ belongs to the approximate point spectrum of $A$ . To do this, we will use the following lemma:

Lemma 2: Let $A : V \rightarrow V$ be a bounded operator on a complex Banach space $V$ . Then, for any sequence $\{x_n\}$ in $V$ such that $\|x_n\| = 1$ for all $n$ , we have

\liminf_{n \rightarrow \infty} \|Ax_n - kx_n\| = 0

if and only if there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that

\lim_{k \rightarrow \infty} \|Ax_{n_k} - kx_{n_k}\| = 0

Proof of Lemma 2: The "only if" part of the lemma follows directly from the definition of the limit inferior. To prove the "if" part, we assume that there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that

\lim_{k \rightarrow \infty} \|Ax_{n_k} - kx_{n_k}\| = 0

We need to show that

\liminf_{n \rightarrow \infty} \|Ax_n - kx_n\| = 0

To do this, we will use the following fact:

Fact 3: Let $\{x_n\}$ be a sequence in a Banach space $V$ . Then, if $\|x_n\| = 1$ for all $n$ , we have

\liminf_{n \rightarrow \infty} \|x_n\| = 1

Proof of Fact 3: This fact follows directly from the definition of the limit inferior.

Using Fact 3, we have

\liminf_{n \rightarrow \infty} \|Ax_n - kx_n\| = \liminf_{k \rightarrow \infty} \|Ax_{n_k} - kx_{n_k}\| = 0

This completes the proof of Lemma 2.

Now, we can complete the proof of Theorem 1. Using Lemma 2, we have

\liminf_{n \rightarrow \infty} \|Ax_n - kx_n\| = 0

if and only if there exists a subsequence $\{x_{n_k}\}$ of $\{x_n\}$ such that

\lim_{k \rightarrow \infty} \|Ax_{n_k} - kx_{n_k}\| = 0

This implies that $k$ belongs to the approximate point spectrum of $A$ . This completes the proof of Theorem 1.

Conclusion

In this article, we have characterised the approximate point spectrum of a bounded operator on a Banach space. We have shown that a complex number $k$ belongs to the approximate point spectrum of a bounded operator $A$ if and only if there exists a sequence $\{x_n\}$ in $V$ such that $\|x_n\| = 1$ for all $n$ , and

\liminf_{n \rightarrow \infty} \|Ax_n - kx_n\| = 0

This result has important implications for the study of spectral theory, and it provides a useful tool for understanding the properties of bounded operators on Banach spaces.

References

[1]: A. Pietsch, "Operator Theory", Birkhäuser, 1980.
[2]: J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces I", Springer, 1977.
[3]: R. G. Douglas, "Banach Algebra Techniques in Operator Theory", Academic Press, 1972.

Introduction

In our previous article, we characterised the approximate point spectrum of a bounded operator on a Banach space. In this article, we will answer some frequently asked questions about this topic, providing further clarification and insights into the properties of the approximate point spectrum.

Q: What is the approximate point spectrum of a bounded operator?

A: The approximate point spectrum of a bounded operator $A$ on a Banach space $V$ is the set of complex numbers $k$ such that there exists a sequence $\{x_n\}$ in $V$ with $\|x_n\| = 1$ for all $n$ and

\liminf_{n \rightarrow \infty} \|Ax_n - kx_n\| = 0

Q: How does the approximate point spectrum relate to the spectrum of a bounded operator?

A: The approximate point spectrum of a bounded operator $A$ is included in the spectrum of $A$ . In other words, if $k$ is in the approximate point spectrum of $A$ , then $k$ is also in the spectrum of $A$ .

Q: What are some properties of the approximate point spectrum of a bounded operator?

A: The approximate point spectrum of a bounded operator $A$ has several interesting properties, including:

Non-emptyness: The approximate point spectrum of a bounded operator is always non-empty.
Closedness: The approximate point spectrum of a bounded operator is a closed set in the complex plane.
Invariance under similarity: The approximate point spectrum of a bounded operator is invariant under similarity transformations.

Q: How can we characterise the approximate point spectrum of a bounded operator?

A: We can characterise the approximate point spectrum of a bounded operator $A$ by using the following theorem:

\liminf_{n \rightarrow \infty} \|Ax_n - kx_n\| = 0

Q: What are some applications of the approximate point spectrum of a bounded operator?

A: The approximate point spectrum of a bounded operator has several applications in functional analysis and operator theory, including:

Spectral theory: The approximate point spectrum of a bounded operator is an important tool in spectral theory, as it provides information about the spectral properties of the operator.
Operator algebras: The approximate point spectrum of a bounded operator is used in the study of operator algebras, which are algebraic structures that arise in the study of operators on Banach spaces.
Functional analysis: The approximate point spectrum of a bounded operator is used in the study of functional analysis, which is a branch of mathematics that deals with the study of vector spaces and linear operators.

Q: What are some open problems related to the approximate point spectrum of a bounded operator?

A: There are several open problems related to the approximate point spectrum of a bounded operator, including:

Characterisation of the approximate point spectrum: Can we characterise the approximate point spectrum of a bounded operator in terms of other spectral properties of the operator?
Relationship between the approximate point spectrum and the spectrum: What is the relationship between the approximate point spectrum and the spectrum of a bounded operator?
Applications of the approximate point spectrum: Are there any new applications of the approximate point spectrum of a bounded operator that have not been explored yet?

Conclusion

In this article, we have answered some frequently asked questions about the approximate point spectrum of a bounded operator on a Banach space. We have provided further clarification and insights into the properties of the approximate point spectrum, and we have discussed some of the applications and open problems related to this topic.

References

[1]: A. Pietsch, "Operator Theory", Birkhäuser, 1980.
[2]: J. Lindenstrauss and L. Tzafriri, "Classical Banach Spaces I", Springer, 1977.
[3]: R. G. Douglas, "Banach Algebra Techniques in Operator Theory", Academic Press, 1972.