In
mathematics, in the field of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
, the Cotlar–Stein almost orthogonality lemma is named after mathematicians
Mischa Cotlar
250px, Mischa Cotlar in 1964
Mischa Cotlar (1913, Sarny, Russian Empire – January 16, 2007, Buenos Aires, Argentina) was a mathematician who started his scientific career in Uruguay and worked most of his life on it in Argentina and Venezuela.
...
and
Elias Stein
Elias Menachem Stein (January 13, 1931 – December 23, 2018) was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathematics, Emeritus, at Princeton University, whe ...
. It may be used to obtain information on the
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introd ...
on an operator, acting from one
Hilbert space into another
when the operator can be decomposed into ''almost orthogonal'' pieces.
The original version of this lemma
(for
self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a st ...
and mutually commuting operators)
was proved by Mischa Cotlar in 1955 and allowed him to conclude that the
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the functi ...
is a
continuous linear operator In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces.
An operator between two normed spaces is a bounded linear op ...
in
without using the
Fourier transform.
A more general version was proved by Elias Stein.
Cotlar–Stein almost orthogonality lemma
Let
be two
Hilbert spaces
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
.
Consider a family of operators
,
,
with each
a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
from
to
.
Denote
:
The family of operators
,
is ''almost orthogonal'' if
:
The Cotlar–Stein lemma states that if
are almost orthogonal,
then the series
converges in the
strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
,
and that
:
Proof
If ''R''
1, ..., ''R''
''n'' is a finite collection of bounded operators, then
:
So under the hypotheses of the lemma,
:
It follows that
:
and that
:
Hence the partial sums
:
form a
Cauchy sequence
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
.
The sum is therefore absolutely convergent with limit satisfying the stated inequality.
To prove the inequality above set
:
with , ''a''
''ij'', ≤ 1 chosen so that
:
Then
:
Hence
:
Taking 2''m''th roots and letting ''m'' tend to ∞,
:
which immediately implies the inequality.
Generalization
There is a generalization of the Cotlar–Stein lemma with sums replaced by integrals.
Let ''X'' be a
locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
and μ a Borel measure on ''X''. Let ''T''(''x'') be a map from ''X'' into bounded operators from ''E'' to ''F'' which is uniformly bounded and continuous in the strong operator topology. If
:
are finite, then the function ''T''(''x'')''v'' is integrable for each ''v'' in ''E'' with
:
The result can be proved by replacing sums by integrals in the previous proof or by using Riemann sums to approximate the integrals.
Example
Here is an example of an ''orthogonal'' family of operators. Consider the inifite-dimensional matrices
:
and also
:
Then
for each
,
hence the series
does not converge in the
uniform operator topology In the mathematical field of functional analysis there are several standard topologies which are given to the algebra of bounded linear operators on a Banach space .
Introduction
Let (T_n)_ be a sequence of linear operators on the Banach space ...
.
Yet, since
and
for
,
the Cotlar–Stein almost orthogonality lemma tells us that
:
converges in the
strong operator topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ...
and is bounded by 1.
Notes
References
*
*
*
*
{{DEFAULTSORT:Cotlar-Stein lemma
Hilbert spaces
Harmonic analysis
Operator theory
Inequalities
Theorems in functional analysis
Lemmas in analysis