A Characterization For Closed Range For An Operator

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Introduction

In the realm of functional analysis, operators play a crucial role in understanding the behavior of linear transformations between normed linear spaces. One of the fundamental properties of an operator is its range, which is the set of all possible output values. In this article, we will delve into the characterization of a closed range for an operator, specifically focusing on the condition that the range of a bounded linear transformation is closed if and only if there exists a positive constant M such that a certain inequality holds.

Preliminaries

Before we dive into the main result, let's establish some necessary background and notation. Let X and Y be normed linear spaces, and let T be a bounded linear transformation from X to Y. The range of T, denoted by R(T), is the set of all possible output values, i.e., R(T) = {Tx | x ∈ X}. The null space of T, denoted by N(T), is the set of all input values that map to the zero vector, i.e., N(T) = {x ∈ X | Tx = 0}. The quotient space X/N(T) is the set of all equivalence classes of X modulo N(T), where two elements x and y are considered equivalent if x - y ∈ N(T).

The Main Result

The main result we will discuss is the characterization of a closed range for an operator. Specifically, we will show that the range of a bounded linear transformation T is closed if and only if there exists a positive constant M such that the following inequality holds:

∥x + N(T)∥ ≤ M ∥Tx∥

for all x ∈ X.

Proof of the Main Result

To prove the main result, we will first assume that the range of T is closed and show that there exists a positive constant M such that the inequality holds. Then, we will assume that there exists a positive constant M such that the inequality holds and show that the range of T is closed.

Proof of (⇒)

Assume that the range of T is closed. We need to show that there exists a positive constant M such that the inequality holds. Let x ∈ X be arbitrary, and let y ∈ X be such that Tx = Ty. Then, we have:

∥x + N(T)∥ = ∥(x - y) + y + N(T)∥ = ∥(x - y) + N(T)∥ ≤ ∥x - y∥ + ∥y + N(T)∥

Since the range of T is closed, we have that y + N(T) ∈ R(T). Therefore, there exists z ∈ X such that y + N(T) = Tz. Then, we have:

∥x - y∥ = ∥(x - z) + (z - y)∥ ≤ ∥x - z∥ + ∥z - y∥ = ∥x - z∥ + ∥Tz - Ty∥ = ∥x - z∥ + ∥T(x - z)∥

Since T is bounded, there exists a positive constant M such that ∥T(x - z)∥ ≤ M ∥x - z∥. Therefore, we have:

∥x - y∥ ≤ M ∥x - z∥

Substituting this into the previous inequality, we get:

∥x + N(T)∥ ≤ (M + 1) ∥x - z∥

Since z ∈ X is arbitrary, we can choose z = x. Then, we have:

∥x + N(T)∥ ≤ (M + 1) ∥x - x∥ = (M + 1) ∥0∥ = 0

Therefore, we have shown that there exists a positive constant M such that the inequality holds.

Proof of (⇐)

Assume that there exists a positive constant M such that the inequality holds. We need to show that the range of T is closed. Let y ∈ R(T) be arbitrary, and let x ∈ X be such that Tx = y. Then, we have:

∥x + N(T)∥ = ∥(x - x) + x + N(T)∥ = ∥x + N(T)∥ ≤ M ∥Tx∥ = M ∥y∥

Since y ∈ R(T) is arbitrary, we have that y + N(T) ∈ R(T). Therefore, the range of T is closed.

Conclusion

In this article, we have discussed the characterization of a closed range for an operator. Specifically, we have shown that the range of a bounded linear transformation T is closed if and only if there exists a positive constant M such that the inequality ∥x + N(T)∥ ≤ M ∥Tx∥ holds for all x ∈ X. This result has important implications for the study of operators and their properties.

References

  • [1] Rudin, W. (1973). Functional Analysis. McGraw-Hill.
  • [2] Yosida, K. (1980). Functional Analysis. Springer-Verlag.
  • [3] Dunford, N., & Schwartz, J. T. (1958). Linear Operators. Part I: General Theory. Interscience Publishers.

Further Reading

For further reading on the topic of operators and their properties, we recommend the following resources:

  • [1] "Operator Theory" by Michael A. Dritschel and Michael A. Dritschel
  • [2] "Functional Analysis" by Walter Rudin
  • [3] "Linear Operators" by Nelson Dunford and Jacob T. Schwartz

Introduction

In our previous article, we discussed the characterization of a closed range for an operator. Specifically, we showed that the range of a bounded linear transformation T is closed if and only if there exists a positive constant M such that the inequality ∥x + N(T)∥ ≤ M ∥Tx∥ holds for all x ∈ X. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the significance of the range of an operator being closed?

A: The range of an operator being closed is significant because it implies that the operator is surjective, meaning that it maps the entire domain onto the entire codomain. This is a desirable property in many applications, such as in the study of differential equations and integral equations.

Q: What is the relationship between the range of an operator and its null space?

A: The range of an operator and its null space are related in the sense that the range is the set of all possible output values, while the null space is the set of all input values that map to the zero vector. In other words, the range is the set of all possible solutions to the equation Tx = y, while the null space is the set of all possible solutions to the equation Tx = 0.

Q: How does the inequality ∥x + N(T)∥ ≤ M ∥Tx∥ relate to the range of an operator?

A: The inequality ∥x + N(T)∥ ≤ M ∥Tx∥ relates to the range of an operator in the sense that it provides a bound on the norm of the input value x in terms of the norm of the output value Tx. This is a useful tool for studying the properties of operators and their ranges.

Q: Can you provide an example of an operator whose range is not closed?

A: Yes, consider the operator T: ℝ → ℝ defined by T(x) = x^2. The range of this operator is not closed because it does not contain the point 0, even though 0 is the limit of the sequence {1/n^2}.

Q: Can you provide an example of an operator whose range is closed?

A: Yes, consider the operator T: ℝ → ℝ defined by T(x) = x. The range of this operator is closed because it contains all real numbers.

Q: How does the characterization of a closed range for an operator relate to other areas of mathematics?

A: The characterization of a closed range for an operator relates to other areas of mathematics, such as functional analysis, operator theory, and differential equations. It is a fundamental result in the study of operators and their properties, and has important implications for many applications.

Q: What are some potential applications of the characterization of a closed range for an operator?

A: Some potential applications of the characterization of a closed range for an operator include:

  • Studying the properties of differential equations and integral equations
  • Analyzing the behavior of linear systems and control theory
  • Developing new methods for solving linear equations and optimization problems
  • Understanding the properties of operators in quantum mechanics and other areas of physics

Conclusion

In this article, we have answered some frequently asked questions related to the characterization of a closed range for an operator. We hope that this article has provided a useful resource for those interested in this topic.

References

  • [1] Rudin, W. (1973). Functional Analysis. McGraw-Hill.
  • [2] Yosida, K. (1980). Functional Analysis. Springer-Verlag.
  • [3] Dunford, N., & Schwartz, J. T. (1958). Linear Operators. Part I: General Theory. Interscience Publishers.

Further Reading

For further reading on the topic of operators and their properties, we recommend the following resources:

  • [1] "Operator Theory" by Michael A. Dritschel and Michael A. Dritschel
  • [2] "Functional Analysis" by Walter Rudin
  • [3] "Linear Operators" by Nelson Dunford and Jacob T. Schwartz