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Chapter 6: Problem 7

Show that every bounded operator can be written as a linear combination of two self-adjoint operators. Furthermore, show that every bounded self-adjoint operator can be written as a linear combination of two unitary operators. (Hint: \(x \pm \mathrm{i} \sqrt{1-x^{2}}\) has absolute value one for \(x \in\) \([-1,1]\).)

Short Answer

Expert verified
A bounded operator is a combination of two self-adjoint operators; a self-adjoint operator is a combination of two unitary operators.

Step by step solution

01

Understanding the Problem

The problem asks us to express a bounded operator as a linear combination of two self-adjoint operators and to express a bounded self-adjoint operator as a linear combination of two unitary operators. The hint involves properties of specific expressions, which suggests a connection between the bounded and self-adjoint operators with unitary ones.
02

Expressing a Bounded Operator

Consider any bounded operator \(A\) on a Hilbert space. We can express \(A\) as: \(A = \text{Re}(A) + i\text{Im}(A)\), where \(\text{Re}(A) = \frac{A + A^*}{2}\) and \(\text{Im}(A) = \frac{A - A^*}{2i}\). Both \(\text{Re}(A)\) and \(\text{Im}(A)\) are self-adjoint. Thus, \(A\) is a linear combination of two self-adjoint operators.
03

Properties of Self-Adjoint Operators

Given a self-adjoint operator \(S\) with spectrum in \([-1, 1]\), we can create unitary operators using the hint provided. Consider the operator \(U = S + i\sqrt{1-S^2}\), which is unitary because its eigenvalues have an absolute value of one.
04

Expressing a Self-Adjoint Operator as Unitaries

Now, we express the self-adjoint operator \(S\) using the unitary operator \(U\). Notice that \(U + U^* = 2S\). Thus, \(S = \frac{1}{2}(U + U^*)\), which shows that \(S\) as a self-adjoint operator can be expressed as a linear combination of the unitary operators \(U\) and \(U^*\).
05

Conclusion

From these constructions, we see any bounded operator is a combination of two self-adjoint operators and any bounded self-adjoint operator can be a combination of two unitary operators, thereby verifying the statements in the problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Self-Adjoint Operators
Self-adjoint operators are a fundamental concept in functional analysis, especially within the context of a Hilbert space. An operator is termed self-adjoint if it equals its own adjoint, meaning for an operator \( A \) acting on a Hilbert space, \( A = A^* \). This characteristic implies several remarkable properties:
  • Reality of Eigenvalues: Every self-adjoint operator has real eigenvalues. This is vital because it mirrors the properties of real symmetric matrices but extends these over potentially infinite-dimensional spaces.
  • Orthogonality of Eigenvectors: Eigenvectors corresponding to different eigenvalues of a self-adjoint operator are orthogonal. This property allows the operator to be decomposed into simpler, more analyzable pieces.
  • Role in Quantum Mechanics: Self-adjoint operators are used to represent observable quantities in quantum mechanics. This association underscores their importance as measurable, physically significant quantities.
Understanding self-adjoint operators can assist in dealing with complex vectors and transformations in Hilbert spaces.
Unitary Operators
Unitary operators are central to the study of transformations on a Hilbert space because they preserve inner products, effectively maintaining the length and angle of vectors.
  • Definition: An operator \( U \) is unitary if \( U^*U = UU^* = I \), where \( I \) is the identity operator. This condition means that the adjoint of \( U \) is also its inverse, maintaining its operations across the space without distortion.
  • Preservation Properties: They preserve norms and inner products, hence unitary transformations are inherently isometries of the space. This makes them vital in quantum mechanics and signal processing where preserving structure is crucial.
  • Eigenvalue Characteristics: Unitary operators have eigenvalues with an absolute value of one. This stems from the preservation of length after transformation, aligning with the unit circle in the complex plane.
By understanding these operators, we can elegantly transition between various states or configurations without altering the fundamental nature of data or states being processed.
Hilbert Space
A Hilbert space is a complete vector space equipped with an inner product, allowing distance measurements and angles between vectors. This forms a natural extension of Euclidean space, encompassing even infinite dimensions while retaining integral properties of finite-dimensional spaces.
  • Completeness: This is a key attribute, meaning every Cauchy sequence in the space has a limit within the space. It ensures the space can accommodate any limiting behavior of sequences, which is crucial for analysis and functional operations.
  • Inner Product Space: The presence of an inner product allows for the definition of angle and length, facilitating concepts like orthogonality and projection, vital for many mathematical and physical applications.
  • Applications: In quantum mechanics, Hilbert spaces serve as the backbone for state spaces where wave functions reside. Their utility extends beyond into signal processing, statistics, and data science, providing a sophisticated framework for diverse analyses.
A deep grasp of Hilbert space theory can pave the way for exploring more complex phenomena and operations on vector spaces with greater dimensionality.
Linear Combination
Linear combinations are at the heart of understanding how vectors in a space can be combined to form new vectors, which is a fundamental operation in linear algebra.
  • Definition: A linear combination involves summing scaled (or multiplied by scalars) vectors. For vectors \( v_1, v_2, \, \ldots, \, v_n \) and scalars \( a_1, a_2, \, \ldots, \, a_n \), a linear combination is expressed as \( a_1v_1 + a_2v_2 + \ldots + a_nv_n \).
  • Span and Basis: The notion of a linear combination is tightly bound to the concept of span, which considers all possible linear combinations of a set of vectors. If this spans a space, these vectors can form a basis, key in dimension determination.
  • Relevance in Operators: By expressing complex operators in terms of simpler, often orthogonal, operators through linear combinations, one can greatly simplify analysis and computation.
Understanding linear combinations can give insights into solving linear equations, transforming spaces, and optimizing conditions across various mathematical and physical domains.

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Most popular questions from this chapter

Show that \(A=-\frac{d^{2}}{d x^{2}}+q(x), \mathcal{D}(A)=H^{2}(\mathbb{R})\) is self-adjoint if \(q \in L^{\infty}(\mathbb{R})\). Show that if \(-u^{\prime \prime}(x)+q(x) u(x)=z u(x)\) has a solution for which \(u\) and \(u^{\prime}\) are bounded near \(+\infty(\) or \(-\infty)\) but \(u\) is not square integrable near \(+\infty(\) or \(-\infty)\), then \(z \in \sigma_{\text {ess }}(A)\). (Hint: Use \(u\) to construct a Weyl sequence by restricting it to a compact set. Now modify your construction to get a singular Weyl sequence by observing that functions with disjoint support are orthogonal.)

Suppose \(K, K_{0}\) are bounded self-adjoint operators. Show that if \(-K_{0} \leq K \leq K_{0}\), then \(\|K\|_{p} \leq\left\|K_{0}\right\|_{p}\). (Hint: Corollary 4.13.)

An operator of the form \(K: \ell^{2}(\mathrm{~N}) \rightarrow \ell^{2}(\mathrm{~N}), f(n) \rightarrow\) \(\sum_{j \in \mathbb{N}} k(n+j) f(j)\) is called a Hankel operator. \- Show that \(K\) is Hilbert-Schmidt if and only if \(\sum_{j \in \mathbb{N}} j|k(j)|^{2}<\infty\) and this number equals \(\|K\|_{2}\). \- Show that \(K\) is Hilbert-Schmidt with \(\|K\|_{2} \leq\|c\|_{1}\) if \(|k(n)| \leq c(n)\), where \(c(n)\) is decreasing and summable.

Let \(A\) and \(B\) be self-adjoint operators. Suppose \(B\) is relatively bounded with respect to \(A\) and \(A+B\) is self-adjoint. Show that if \(|B|^{1 / 2} R_{A}(z)\) is Hilbert-Schmidt for one \(z \in \rho(A)\), then this is true for all \(z \in \rho(A) .\) Moreover, \(|B|^{1 / 2} R_{A+B}(z)\) is also Hilbert- Schmidt and \(R_{A+B}(z)-\) \(R_{A}(z)\) is trace class.

Suppose \(A_{n} \rightarrow A\) in the norm resolvent sense and let \(A\) be bounded. Show that \(A_{n}\) are eventually bounded and \(A_{n} \rightarrow A\) in norm. (Hint: First show \(\left\|A_{n}\right\| \rightarrow\|A\|\) and conclude that \(R_{A}(\mathrm{i})\) and \(R_{A_{n}}(\mathrm{i})\) are eventually bi- Lipschitz uniformly in \(n\). Now use the second resolvent formula.)
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