Toeplitz Operators

Preliminaries and the Hardy Space

Let’s establish the setting for our discussion.

• We use the normalized arc length measure for .
• We denote the Lebesgue spaces as .
• The functions for form an orthonormal basis for the Hilbert space .

Hardy Space

The Hardy space is the subspace of consisting of functions whose negative Fourier coefficients vanish: is a closed subspace of and thus is a Hilbert space in its own right. The set forms an orthonormal basis for .

Multiplication and Toeplitz Operators

We can define operators on using functions from .

Multiplication Operator

For any , the multiplication operator is defined by:

Proposition

The multiplication operator is a bounded linear operator with the following properties:

• .
• The map is a -homomorphism onto its image. Specifically, , , and .
• is invertible if and only if is invertible in .

Using the multiplication operator and the projection onto the Hardy space, we can define the Toeplitz operator.

Toeplitz Operator

Let be the orthogonal projection. For any , the Toeplitz operator is defined by:

Proposition

We have the following properties of Toeplitz operators:

• Boundedness: is a bounded operator on with .
• Linearity: The map is linear.
• Adjoint: .
• Non-multiplicativity: In general, .
• If is invertible, then is invertible, but in general.
• The spectrum satisfies .

The structure of Toeplitz operators becomes clear when we examine their matrix representation with respect to the orthonormal basis of .

Toeplitz Matrix

For , the -th matrix entry of is: This means the matrix of has constant entries along its diagonals. Such a matrix is called a Toeplitz matrix:

e.g. Let . Then and all other Fourier coefficients are zero. The operator is the unilateral shift operator , which acts as . Its adjoint, , is the backward shift.

The algebraic structure of Toeplitz operators is rich and subtle.

Compactness Condition

A Toeplitz operator is compact if and only if its symbol is identically zero.

Commutator Property

For continuous symbols , the commutator is compact: This means that Toeplitz operators with continuous symbols commute modulo the compacts.

Proof We can prove this by induction on the Fourier modes of the symbols.

1. Base Case: For , we show that is a rank-one operator. For : So, . Then . The difference is the rank-one operator , which is compact.
2. Inductive Step: Assume is compact. Then we can write: Since the compact operators form an ideal, the sum is compact. By linearity and density of trigonometric polynomials in , the result holds for all .

The Toeplitz Algebra

Toeplitz Algebra

The Toeplitz algebra, denoted , is the unital C*-algebra generated by all Toeplitz operators with continuous symbols: with unit .

A remarkable result is that this algebra contains all compact operators and is generated by a single element.

Proposition

The Toeplitz algebra contains the ideal of compact operators, . Furthermore, the algebra is generated by the unilateral shift: .

Proof The operator is the projection onto the span of . This is a rank-one operator in . From this, we can construct matrix units , which are also in . The linear span of these matrix units is the set of finite-rank operators. Since is closed, it contains the closure of the finite-rank operators, which is precisely .

The structure of the Toeplitz algebra is completely characterized by the following fundamental theorem.

Structure of the Toeplitz Algebra

The C*-algebra of compact operators is a closed ideal in the Toeplitz algebra . The quotient algebra is -isomorphic to the algebra of continuous functions on the circle, . This relationship is summarized by the short exact sequence: The symbol map is a -homomorphism defined by , with .

Proof It suffices to prove that , is a -isomorphism. is linear and preserves adjoints. Since , we have that is multiplicative, so is a -homomorphism. Since , is surjective. If , then is compact, so , hence is injective.