Question

Show that every Hilbert-Schmidt operator \(T\) on a Hilbert space \(H\) is compact. Find a compact operator that is not a Hilbert-Schmidt operator.

Short answer

A Hilbert-Schmidt operator is compact because the sequence of images of the orthonormal basis converges to zero. The identity operator on an infinite-dimensional H is a compact operator that is not Hilbert-Schmidt.

Step-by-step solution

Definitions and overview

First, recall the definition of a Hilbert-Schmidt operator. An operator T on a Hilbert space H is Hilbert-Schmidt if for an orthonormal basis \( \{ e_n \} \) of H, the sum \( \sum_{n} \| T e_n \|^2 < \infty \). A compact operator is one that maps bounded sets to relatively compact sets (i.e., the closure of the image is compact).

Show Hilbert-Schmidt operator is compact

Consider an orthonormal basis \( \{ e_n \} \) of H. For a Hilbert-Schmidt operator T, the sequence \( T e_n \) is in l^2, meaning \( \sum_{n} \| T e_n \|^2 < \infty \). This implies that \( T e_n \rightarrow 0 \) as \( n \rightarrow \infty \). Now, for any bounded sequence \( x_k \) in H, the sequence \( T x_k \) lies in the closed linear span of \( \{ T e_n \} \) which is finite-dimensional. Hence, T is a compact operator.

Counterexample for compact operator not Hilbert-Schmidt

Consider the identity operator on an infinite-dimensional Hilbert space H. The identity operator is not Hilbert-Schmidt because \( \| I e_n \| = 1 \) for all \( n \), making the series \( \sum_{n} \| I e_n \|^2 = \sum_{n} 1 = \infty \). However, the identity operator is not compact because it does not map bounded sets to relatively compact sets.

Key concepts

Compact Operators

Compact operators are an important class in functional analysis that have many nice properties. An operator \( T \) on a Hilbert space \( H \) is compact if it maps bounded sets to relatively compact sets. This means that if you take any bounded set in the Hilbert space, its image under the operator will have a closure that is compact.
An example might help: Think of a function that takes points from a big circle and maps them inside a smaller circle. Even though the initial set (big circle) is bounded, the image set (smaller circle) is not only bounded but also compact because it is closed and bounded.
Key features of compact operators include:

• They generalize the idea of matrices to infinite dimensions.
• They have eigenvalues that tend to zero.
• They can simplify complex problems into more manageable pieces.

Understanding compact operators helps in many areas of mathematics and physics.

Hilbert Space

A Hilbert space is a complete inner-product space, a generalization of Euclidean space. Imagine a space with an infinite basis, where each pair of vectors has an inner product (like a dot product).
The inner product in a Hilbert space allows you to measure angles and lengths, making it a powerful tool for mathematical analysis. \( H \) is a common notation for Hilbert space.
Key properties:

• Every Cauchy sequence in a Hilbert space converges.
• An essential structure for quantum mechanics and signal processing.
• Supports an orthonormal basis, where vectors are at right angles to each other and have unit length.

Hilbert spaces provide the framework for many modern mathematical theories and physical applications.

Orthonormal Basis

An orthonormal basis in a Hilbert space is a set of vectors that are both orthogonal and normalized. This means each pair of vectors is at a right angle, and each vector has a length of one.
Having an orthonormal basis is extremely useful because it allows for decomposing any vector in the space into a unique combination of these basis vectors. Representing functions and signals in terms of their orthonormal basis elements can simplify many problems.
To break it down:

• Orthogonal: Vectors are at right angles to each other.
• Normalized: Vectors have unit length.
• Basis: You can express every vector in the space uniquely using these vectors.

Understanding orthonormal bases helps in analyzing and simplifying problems in various fields such as physics, engineering, and computer science.

Bounded Sets

A set in a Hilbert space is bounded if there is a limit to how far the elements of the set can go from a reference point, usually the origin. You can think of it like a circle or sphere with a fixed radius; nothing inside can go beyond that distance.
Why bounded sets are essential:

• They provide control over the behavior of sequences and functions.
• Important in determining the compactness of an operator.
• Helps in proving various theorems in analysis and topology.

In any analysis or application, knowing that a set is bounded ensures that you won’t encounter values or vectors that 'explode' into infinity, making your work much more manageable and predictable.