# On Truncated Toeplitz Operators

Introductory notes on the algebraic properties of truncated Toeplitz operators.

Our setting is the open unit disk, D and T the unit circle, in the complex plane, C. By H² is meant the standard Hardy space, the Hilbert space of holomorphic functions in D having square-summable Taylor coefficients at the origin. As usual, H² will be identified with its space of boundary functions, the subspace of L² (of normalized Lebesgue measure m on T) consisting of the functions whose Fourier coefficients with negative indices vanish.

A Toeplitz operator is the compression to H² of a multiplication operator on L². The operators of the paper’s title are compressions of multiplication operators to proper invariant subspaces of the backward shift operator on H². An effort has been exerted to make the paper reasonably self-contained.

Some preparation is needed prior to precise definitions.

We let P denote the

orthogonal projection on L²with range H². The operator P is given explicitly as a Cauchy integral:

We shall need to deal with certain unbounded Toeplitz operators.

The operator T_ϕ, the Toeplitz operator on H² with symbol ϕ, is defined by

We let S denote the unilateral shift operator on H². Its adjoint, the backward

shift, is given by

Clearly, a truncated Toeplitz operator does not have a unique symbol.

The compression of S to K_u will be denoted by S_u. Its adjoint, S*_u, is the

restriction of S* to K_u. The operators S_u and S*_u are the truncated Toeplitz operators with symbols z and z, respectively.

The next section contains most of the needed background on the spaces.

# Background Materials;

The bulk of it can be found in standard sources;

The model space, as is well known, carries a natural conjugation, an antiunitary involution C, defined by

It preserves the model spaces.

An operator A on model space K_u is called C-symmetric if CAC = A*. S. R. Garcia and M. Putinar study the notion of C-symmetry in the abstract sense. They give many examples, including our truncated Toeplitz operators (at least those with bounded symbols). The following result is essentially theirs.

Let T(K_u) denote the space of all bounded truncated Toeplitz operators defined on the model space K_u.

We have the following results which are called defect operators or rank one operators in terms of reproducing kernel Hilbert spaces.

# Condition for A_phi=0;

In this section will characterize the zero truncated Toeplitz operators whose related with its symbol.

The symbol of the truncated Toeplitz operator is not unique. The following corollary shows this…

# Characterizations of Truncated Toeplitz Operators;

The bulk of the proof will be accomplished in two lemmas.

The following result is that the space of truncated Toeplitz opertaors is closed in the weak operator topology.

# Characterization by Shift Invariance;

Given a bounded operator A on model space K_u, we let Q_A denote the associated quadratic form on K_u,

We shall say that A is shift invariant if

whenever f and Sf are in K_u.

If this happens then, by the polarization identity, we also have

whenever f,g, Sf,Sg are belong to the model space K_u.

Truncated Toeplitz operators can be characterize by using the following theorem.

# Crofoot’s Transforms

The following theorem shows that J_w is an isometry between model spaces.

The adjoint of the Crofoot transform is given by

The Crofoot transform of a truncated Toeplitz operator is a

truncated Toeplitz operator on another model space.

The following lemmas can be used to prove the above theorem.