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Chapter 1
Introduction
1.1 Preliminary notions and notations
In this section we collect some notions and facts of the theory of infinite matrices, the theory of analytic functions on the disk and the circle, of vector-valued integration theory and of geometry of the disk etc.
1.1.1 Infinite matrices
For an infinite matrix A = (aij), and an integer k we denote by Ak the matrix whose entries a′i,j are given by
Then Ak will be called the kth-diagonal matrix associated to A.
Sometimes we use also the notation a(i, j) for the entries of the matrix A.
An important notion in the theory of matrices is the
Schur product. Let
A = (
aij)
i,j and
B = (
bij)
i,j be two infinite matrices. Then the
Schur product C = (
cij)
i,j of
A and B, denoted by
A * B, has the entries
cij =
aijbij for all
i, j ∈
.
An infinite matrix A such that A * B ∈ Y for all B ∈ X, where X, Y are Banach spaces of infinite matrices, is called a Schur multiplier from X into Y, and the space of all Schur multipliers from X into Y, endowed with the natural norm
is denoted by (X, Y).
In the case X = Y = B(ℓ2), where B(ℓ2) is the space of all linear and bounded operators on ℓ2, the space (X, Y) is denoted by M(ℓ2) (an explanation of this notation is given later in this section) and a matrix A ∈ M(ℓ2) is simply called a Schur multiplier.
We consider on the interval [0, 1) the Lebesgue measurable infinite matrix valued functions
A(
r). These functions may be regarded as infinite matrix-valued functions defined on the unit disk
D using the correspondence
where
Ak(
r) is the
kth-diagonal of the matrix
A(
r), the preceding sum is a formal one and
t belongs to the torus
.
We may consider fA(r, t), or fA(z), with z = reit, as a matrix valued function, or distribution, or just a formal series.
Such a matrix
A(
r) is called
an analytic matrix if there exists an upper triangular infinite matrix
A such that, for all
r ∈ [0, 1), we have
Ak(
r) =
Akrk, for all
k ∈
.
In what follows we identify the analytic matrices A(r) with their corresponding upper triangular matrices A and call the latter also as analytic matrices.
A special class of infinite matrices is considered often in this book, namely the class of Toeplitz matrices.
Let
A = (
aij)
i,j≥1 be an infinite matrix. If there is a sequence of complex numbers
such that
aij =
aj–i for all
i, j ∈
, then
A is called
a Toeplitz matrix.
