Cotlar–Stein lemma

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The Cotlar–Stein almost orthogonality lemma is a mathematical lemma in the field of functional analysis. It may be used to obtain information on the operator norm 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 and mutually commuting operators) was proved by Mischa Cotlar in 1955[1] and allowed him to conclude that the Hilbert transform is a continuous linear operator in without using the Fourier transform. A more general version was proved by Elias Stein.[2]

Statement of the lemma

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Let   be two Hilbert spaces. Consider a family of operators  ,  , with each   a bounded linear operator 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, and

 

Proof

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If   is a finite collection of bounded operators, then[3]

 

So under the hypotheses of the lemma,

 

It follows that

 

and that

 

Hence, the partial sums

 

form a Cauchy sequence.

The sum is therefore absolutely convergent with the limit satisfying the stated inequality.

To prove the inequality above set

 

with |aij| ≤ 1 chosen so that

 

Then

 

Hence

 

Taking 2mth roots and letting m tend to ∞,

 

which immediately implies the inequality.

Generalization

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The Cotlar-Stein lemma has been generalized, with sums being replaced by integrals.[4][5] Let X be a locally compact space 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 proven by replacing sums with integrals in the previous proof, or by utilizing Riemann sums to approximate the integrals.

Example

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Here is an example of an orthogonal family of operators. Consider the infinite-dimensional matrices.

 

and also

 

Then   for each  , hence the series   does not converge in the uniform operator topology.

Yet, since   and   for  , the Cotlar–Stein almost orthogonality lemma tells us that

 

converges in the strong operator topology and is bounded by 1.

Notes

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  1. ^ Cotlar 1955
  2. ^ Stein 1993
  3. ^ Hörmander 1994
  4. ^ Knapp & Stein 1971
  5. ^ Calderon, Alberto; Vaillancourt, Remi (1971). "On the boundedness of pseudo-differential operators". Journal of the Mathematical Society of Japan. 23 (2): 374–378. doi:10.2969/jmsj/02320374.

References

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  • Cotlar, Mischa (1955), "A combinatorial inequality and its application to L2 spaces", Math. Cuyana, 1: 41–55
  • Hörmander, Lars (1994), Analysis of Partial Differential Operators III: Pseudodifferential Operators (2nd ed.), Springer-Verlag, pp. 165–166, ISBN 978-3-540-49937-4
  • Knapp, Anthony W.; Stein, Elias (1971), "Intertwining operators for semisimple Lie groups", Ann. Math., 93: 489–579, doi:10.2307/1970887, JSTOR 1970887
  • Stein, Elias (1993), Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, ISBN 0-691-03216-5