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Given a linear map $T:H\to H$ on an inner-product space $H$ and a subspace $K\subseteq H$, define the map $T_K = \pi_K T \pi_K^* :K \to K$, where $\pi_K:H\to K$ is the orthogonal projection.

As an important special case, if $H=\mathbb{R}^n$ and $K$ is a coordinate subspace, then with respect to standard bases, $T_K$ is represented by a principal submatrix of the matrix of $T$.

Is there a standard, or at least widely recognized, name for $T_K$ when $K$ is not a coordinate subspace of $\mathbb{R}^n$? The book Matrix Analysis by R. Bhatia calls $T_K$ the compression of $T$ to $K$, but I haven't seen that word used in this way elsewhere (and it's a tricky word to google).

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I don't think I have heard any name other than “compression”, but it's not terribly much used I think, so it is probably fairly useless as a search term. – Harald Hanche-Olsen Jan 5 '10 at 20:09
It's neat to see that this was useful: – Jonas Meyer Jan 20 '10 at 6:43
@Jonas: yes, needing the right terminology for that paper was exactly the reason for the question. So thanks for the helpful answer! – Mark Meckes Jan 20 '10 at 14:05
up vote 6 down vote accepted

The standard name in operator theory is "compression", and its partner in crime is "dilation". I.e., A is a compression of B if and only if B is a dilation of A (although sometimes "dilation" is reserved for cases where the compression respects powers). The Wikipedia entry is not proof, but here it is anyway. As for searches, you'll get some relevant hits from "compression of an operator" with quotes.

Here are some examples.

Some further remarks:

Sz.-Nagy and Foiaș in Harmonic analysis of operators on Hilbert space (1970) use the notation $\text{pr }T$ for the compression of $T$ onto $K$ (see page 10), but apparently without ever giving it a name. The notation is suggestive of "projection", and that is the terminology used by Sarason in "Generalized interpolation in $H^\infty$" (1967).

Lebow goes into more detail on terminology in "A note on normal dilations" (1965), saying in particular that Sz.-Nagy used "projection". In fact, this is the terminology used by Sz.-Nagy in the celebrated appendix to Riesz and Sz.-Nagy's Functional analysis (1955), which in turn refers to Halmos's paper "Normal dilations and extensions of operators" (1950) as the first place where "compression" and "dilation" were used. The terminology "strong compression" may be used when the compression respects powers, and this is the same as saying that $K$ is semi-invariant for $T$ (see Sarason's "On spectral sets having connected complement" (1965)). If $K$ is reducing for $T$, i.e., if both $K$ and $K^\perp$ are invariant subspaces for $T$, then Lebow calls the compression a "reduction".

Dixmier gives some terminology in von Neumann algebras (translated 1981 printing) for the case when the compression is applied to an entire von Neumann algebra of operators, which clashes somewhat with the terminology of Lebow. A von Neumann algebra compressed to the space of a projection in the algebra is called a "reduced" von Neumann algebra (page 19), even though the space is reducing only if the projection is in the center. The compression of a von Neumann algebra onto the space of a projection in the commutant (in which case the compression is a normal $*$-homomorphism) is called an "induction". If $P$ denotes the orthogonal projection you called $\pi_K$, then Dixmier uses the notation $T_K$ or $T_P$ for the compression, but without ever giving a name to the construction for single operators. On the other hand, Jones and Sunder use "reduction" for what Dixmier calls "induction", more in tune with Lebow, on page 21 of Introduction to subfactors (1997).

I stand by my answer that by now "compression" is most standard for single operators, and it is satisfying to find out that we have Halmos to thank for this.

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