In
operator theory, a Toeplitz operator is the
compression of a
multiplication operator
In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,
T_f\varphi(x) = f(x) \varphi (x) \quad
for all in th ...
on the circle to the
Hardy space.
Details
Let ''S''
1 be the circle, with the standard Lebesgue measure, and ''L''
2(''S''
1) be the Hilbert space of square-integrable functions. A bounded measurable function ''g'' on ''S''
1 defines a
multiplication operator
In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function . That is,
T_f\varphi(x) = f(x) \varphi (x) \quad
for all in th ...
''M
g'' on ''L''
2(''S''
1). Let ''P'' be the projection from ''L''
2(''S''
1) onto the
Hardy space ''H''
2. The ''Toeplitz operator with symbol g'' is defined by
:
where " , " means restriction.
A bounded operator on ''H''
2 is Toeplitz if and only if its matrix representation, in the
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
, has constant diagonals.
Theorems
* Theorem: If
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
, then
is
Fredholm if and only if
is not in the set
. If it is Fredholm, its index is minus the winding number of the curve traced out by
with respect to the origin.
For a proof, see . He attributes the theorem to
Mark Krein,
Harold Widom
Harold Widom (September 23, 1932 – January 20, 2021) was an American mathematician best known for his contributions to operator theory and random matrices. He was appointed to the Department of Mathematics at the University of California, Santa ...
, and Allen Devinatz. This can be thought of as an important special case of the
Atiyah-Singer index theorem.
*
Axler-
Chang-
Sarason Sarason is a surname. Notable people with the surname include:
* Donald Sarason (1933–2017), American mathematician
* Leonard Sarason (1925–1994), American composer
*Seymour Sarason (1919–2010), American psychologist
See also
*Sarason interp ...
Theorem: The operator
is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
if and only if
.
Here,
denotes the closed subalgebra of
of analytic functions (functions with vanishing negative Fourier coefficients),