The Convergence Of The Adjoint Operator

Introduction

In the realm of functional analysis, operators play a crucial role in understanding various mathematical structures. The concept of adjoint operators is a fundamental aspect of operator theory, which has far-reaching implications in the study of Hilbert spaces. In this article, we will delve into the convergence of adjoint operators, exploring the relationship between the convergence of a sequence of operators and the convergence of their adjoints.

Background

Let $H$ be a Hilbert space, and let $B(H)$ denote the space of bounded linear operators on $H$. For an operator $A \in B(H)$, the adjoint operator $A^*$ is defined as the unique operator satisfying

$$\langle Ax, y \rangle = \langle x, A^*y \rangle$$

for all $x, y \in H$. The adjoint operator is a crucial tool in operator theory, as it enables us to study the properties of operators in a more abstract and elegant manner.

Convergence of Operators

A sequence of operators $\{A_n\}$ is said to converge in norm to an operator $A$ if

$$\lim_{n \to \infty} \|A_n - A\| = 0$$

where $\| \cdot \|$ denotes the operator norm. This concept of convergence is a fundamental aspect of functional analysis, as it allows us to study the behavior of operators in a more precise and quantitative manner.

Convergence of Adjoint Operators

The question of whether the adjoint operator $A_n^*$ converges in norm to $A^*$ when $A_n$ converges in norm to $A$ is a subtle one. At first glance, it may seem that the answer is affirmative, as the adjoint operator is a continuous function of the operator. However, a closer examination reveals that the situation is more complex.

Counterexample

To demonstrate that the convergence of $A_n^*$ to $A^*$ is not guaranteed, we present a counterexample. Let $H = \mathbb{C}^2$ and define the operators $A_n$ and $A$ as follows:

$$A_n = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$$

$$A = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$$

It is clear that $\|A_n - A\| = 1$ for all $n$, so the sequence $\{A_n\}$ does not converge in norm to $A$. However, we can compute the adjoint operators as follows:

$$A_n^* = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$$

$$A^* = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$$

In this case, we see that $\|A_n^* - A^*\| = 1$ for all $n$, so the sequence $\{A_n^*\}$ does not converge in norm to $A^*$.

Theorem

Despite the counterexample, we can establish a partial result on the convergence of adjoint operators. We have the following theorem:

Theorem 1. Let $\{A_n\}$ be a sequence of operators in $B(H)$ that converges in norm to an operator $A \in B(H)$. Then, for any $x, y \in H$, we have

$$\lim_{n \to \infty} \langle A_n^*x, y \rangle = \langle A^*x, y \rangle$$

Proof. Let $x, y \in H$ be arbitrary. Then, we have

$$\langle A_n^*x, y \rangle = \langle x, A_ny \rangle$$

Since $\{A_n\}$ converges in norm to $A$, we have

$$\lim_{n \to \infty} \langle x, A_ny \rangle = \langle x, Ay \rangle = \langle A^*x, y \rangle$$

This completes the proof.

Corollary

As a corollary to Theorem 1, we have the following result:

Corollary 1. Let $\{A_n\}$ be a sequence of operators in $B(H)$ that converges in norm to an operator $A \in B(H)$. Then, for any $x \in H$, we have

$$\lim_{n \to \infty} \|A_n^*x\| = \|A^*x\|$$

Proof. Let $x \in H$ be arbitrary. Then, we have

$$\|A_n^*x\| = \sup_{\|y\| = 1} |\langle A_n^*x, y \rangle|$$

Since $\{A_n\}$ converges in norm to $A$, we have

$$\lim_{n \to \infty} \langle A_n^*x, y \rangle = \langle A^*x, y \rangle$$

for all $y \in H$ with $\|y\| = 1$. Therefore, we have

$$\lim_{n \to \infty} \|A_n^*x\| = \sup_{\|y\| = 1} |\langle A^*x, y \rangle| = \|A^*x\|$$

This completes the proof.

Conclusion

In conclusion, we have explored the convergence of adjoint operators, establishing a partial result on the convergence of adjoint operators. While the convergence of $A_n^*$ to $A^*$ is not guaranteed in general, we have shown that the adjoint operator $A_n^*$ converges in norm to $A^*$ for any $x \in H$. This result has far-reaching implications in the study of Hilbert spaces and operator theory, and we hope that this article has provided a useful contribution to the field.

References

• [1] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, 1983.
• [2] E. C. Lance, Hilbert Spaces: A First Course in the Mathematical Structure of Quantum Mechanics, Cambridge University Press, 1995.
• [3] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, 1972.
  Q&A: The Convergence of Adjoint Operators =====================================================

Introduction

In our previous article, we explored the convergence of adjoint operators, establishing a partial result on the convergence of adjoint operators. However, we also presented a counterexample that showed that the convergence of $A_n^*$ to $A^*$ is not guaranteed in general. In this article, we will address some of the most frequently asked questions about the convergence of adjoint operators, providing a deeper understanding of this complex topic.

Q: What is the relationship between the convergence of $A_n$ and the convergence of $A_n^*$?

A: The convergence of $A_n$ to $A$ does not necessarily imply the convergence of $A_n^*$ to $A^*$. However, we have shown that the adjoint operator $A_n^*$ converges in norm to $A^*$ for any $x \in H$.

Q: Can you provide a more general result on the convergence of adjoint operators?

A: Unfortunately, the answer is no. The convergence of $A_n^*$ to $A^*$ is not guaranteed in general, and the counterexample we presented earlier demonstrates this.

Q: What are some common pitfalls to avoid when working with adjoint operators?

A: One common pitfall is to assume that the convergence of $A_n$ to $A$ implies the convergence of $A_n^*$ to $A^*$. Another pitfall is to overlook the fact that the adjoint operator is not necessarily continuous.

Q: How can I determine whether the adjoint operator $A_n^*$ converges to $A^*$?

A: To determine whether the adjoint operator $A_n^*$ converges to $A^*$, you can use the following steps:

1. Check whether the sequence $\{A_n\}$ converges in norm to $A$.
2. Compute the adjoint operators $A_n^*$ and $A^*$.
3. Check whether the sequence $\{A_n^*\}$ converges in norm to $A^*$.

Q: What are some applications of the convergence of adjoint operators?

A: The convergence of adjoint operators has far-reaching implications in various fields, including:

• Quantum mechanics: The convergence of adjoint operators is crucial in the study of quantum systems, where it is used to describe the behavior of particles and systems.
• Signal processing: The convergence of adjoint operators is used in signal processing to analyze and manipulate signals.
• Control theory: The convergence of adjoint operators is used in control theory to design and analyze control systems.

Q: Can you provide some examples of operators for which the adjoint operator converges?

A: Yes, here are some examples of operators for which the adjoint operator converges:

• Shift operators: The shift operator $S$ on the Hilbert space $\ell^2$ is defined as $S(x_1, x_2, \ldots) = (0, x_1, x_2, \ldots)$. The adjoint operator $S^*$ is also a shift operator, and it converges to the zero operator.
• Multiplication operators: The multiplication operator $M_f$ on the Hilbert space $L^2$ is defined as $M_f(x) = f(x)x$. The adjoint operator $M_f^*$ is also a multiplication operator, and it converges to the zero operator.
• Compact operators: The compact operator $K$ on the Hilbert space $H$ is defined as $K(x) = \sum_{n=1}^{\infty} \lambda_n \langle x, e_n \rangle e_n$, where $\{e_n\}$ is an orthonormal basis for $H$. The adjoint operator $K^*$ is also a compact operator, and it converges to the zero operator.

Conclusion

In conclusion, the convergence of adjoint operators is a complex and subtle topic that requires careful consideration. While the convergence of $A_n^*$ to $A^*$ is not guaranteed in general, we have shown that the adjoint operator $A_n^*$ converges in norm to $A^*$ for any $x \in H$. We hope that this article has provided a useful contribution to the field and has helped to clarify some of the most frequently asked questions about the convergence of adjoint operators.

References

• [1] R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Academic Press, 1983.
• [2] E. C. Lance, Hilbert Spaces: A First Course in the Mathematical Structure of Quantum Mechanics, Cambridge University Press, 1995.
• [3] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, Academic Press, 1972.