Question

Let \(H\) be a separable Hilbert space. An operator \(T \in \mathcal{B}(H)\) is called a Hilbert-Schmidt operator if there is an orthonormal basis \(\left\\{e_{i}\right\\}\) of \(H\) such that \(\sum\left\|T\left(e_{i}\right)\right\|^{2}<\infty .\) Show that if \(\left\\{f_{i}\right\\}\) is another orthonormal basis of \(H\), then \(\sum\left\|T\left(f_{i}\right)\right\|^{2}=\sum\left\|T\left(e_{i}\right)\right\|^{2}\) The number \(\|T\|_{H S}=\left(\sum\left\|T\left(e_{i}\right)\right\|^{2}\right)^{\frac{1}{2}}\) is called the Hilbert-Schmidt norm of \(T .\) Show that \(\|T\|_{H S} \geq\|T\|\).

Short answer

The sum \(\sum \Vert T(f_i) \Vert^2\) remains the same for any orthonormal basis \(\{f_i\}\) compared to \(\{e_i\}\). Therefore, \(\|T\|_{HS} \geq \|T\|\).

Step-by-step solution

Define Hilbert-Schmidt Operator

A Hilbert-Schmidt operator is an operator \(T \in \mathcal{B}(H)\) such that there exists an orthonormal basis \(\{e_i\}\) of a separable Hilbert space \(H\) and the sum \(\sum \Vert T(e_i) \Vert^2 < \infty\).

Express Given Sum with Different Basis

Given another orthonormal basis \(\{f_i\}\) of \(H\), we aim to show that \(\sum \Vert T(f_i) \Vert^2 = \sum \Vert T(e_i) \Vert^2\) holds.

Use Parseval's Identity

Parseval's identity states that for any \(x \in H\), \(\Vert x \Vert^2 = \sum_i \vert \langle x,e_i \rangle \vert^2 = \sum_i \vert \langle x,f_i \rangle \vert^2\). Apply this to the vector \(T(e_i)\) and \(T(f_i)\).

Parseval's Identity on \(T(e_i)\)

\(\sum_i \Vert T(e_i) \Vert^2 = \sum_{i,j} \vert \langle T(e_i), e_j \rangle \vert^2\) because \(\{e_j\}\) is an orthonormal basis.

Parseval's Identity on \(T(f_i)\)

\(\sum_i \Vert T(f_i) \Vert^2 = \sum_{i,j} \vert \langle T(f_i), f_j \rangle \vert^2\) because \(\{f_j\}\) is an orthonormal basis.

Equivalence of Sum with Different Bases

From the properties of Hilbert spaces and Parseval's identity, both expressions \(\sum \Vert T(e_i) \Vert^2\) and \(\sum \Vert T(f_i) \Vert^2\) are equal since they represent the norm squared of \(T\) applied to different orthonormal bases.

Define Hilbert-Schmidt Norm

The Hilbert-Schmidt norm of \(T\), \(\|T\|_{HS} = \left( \sum_i \Vert T(e_i) \Vert^2 \right)^{1/2}\), is defined for any orthonormal basis \(\{e_i\}\) of \(H\).

Show \(\|T\|_{HS} \geq \|T\|\)

For any \(x \in H\) with \(\Vert x \Vert = 1\), \(T(x)\) can be expressed in terms of an orthonormal basis \(\{e_i\}\). That is, \(T(x) = \sum_i \langle x, e_i \rangle T(e_i)\). Using the norms: \(\Vert T(x) \Vert^2 = \Vert \sum_i \langle x, e_i \rangle T(e_i) \Vert^2 \leq \sum_i (\vert \langle x, e_i \vert^2 \Vert T(e_i) \Vert^2) \leq \|T\|_{HS}^2\).

Conclusion

Since \(\Vert T(x) \Vert \leq \|T\|_{HS}\) for any unit vector \(x\), we have \(\|T\| \leq \|T\|_{HS}\).

Key concepts

Hilbert space

A Hilbert space is a complete inner-product space. It's a generalization of Euclidean space with infinite dimensions.

Think of it as a mathematical playground where vectors can have infinite components, and operations like addition, scalar multiplication, inner product, and completeness still work. In simpler terms, it's a space where sequences of numbers (vectors) approach a limit. This limit is also part of the space.

The beauty of Hilbert spaces lies in their structure. They have properties like inner products that help measure angles and lengths just like in our familiar 3D world.

Orthonormal basis

An orthonormal basis in a Hilbert space is a set of vectors that are both orthogonal (perpendicular) and normalized (have a length of one).

This means each vector in the basis is different from the others, and has a special property: the inner product (or dot product) of different vectors is zero, and the inner product of a vector with itself is one.

Why is this important? Because any vector in the Hilbert space can be written as a combination of these basis vectors.

Imagine you want to describe a complicated vector. Instead of struggling, you break it down into simple, easy-to-understand parts using the orthonormal basis. It simplifies complex problems and is a powerful tool in mathematical analysis.

Parseval's identity

Parseval's identity is a fundamental theorem in Fourier analysis. It helps us equate the total energy (or norm) of a function with the sum of the energies of its Fourier coefficients.

In a Hilbert space, it simply means that if you have an orthonormal basis \( \{e_i\} \), then for any vector \( x \) in the space: \[ \| x \|^2 = \sum_i | \langle x, e_i \rangle |^2. \]

This identity is hugely useful because it tells us that no matter which orthonormal basis we use to represent our vector, the sum of its squared projections remains the same.

Think of it as spread. If you spread butter (the vector) on a slice of bread (the orthonormal basis), the total amount of butter doesn't change.

Hilbert-Schmidt norm

The Hilbert-Schmidt norm is a specific type of norm used to measure the size of an operator. For an operator \( T \) on a Hilbert space with an orthonormal basis \( \{e_i\} \): \[ \| T \|_{HS} = \left( \sum_i \| T(e_i) \|^2 \right)^{1/2}. \]

It gauges how 'big' an operator is by squaring the lengths of its effects on basis vectors, summing these squares, and taking the square root.

One crucial thing here is that this norm is independent of the choice of orthonormal basis. It stays the same regardless of which basis vectors you use.

Moreover, it also directly relates to the usual operator norm (\| T \|), showing that the Hilbert-Schmidt norm is greater than or equal to the operator norm. This provides a comprehensive way to measure operators in infinite-dimensional spaces.