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In operator theory, a bounded operator T: X → Y between normed vector spaces X and Y is said to be a contraction if its operator norm ||T|| ≤ 1. Every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators.

Contractions on a Hilbert space

Consider the special case where T is a contraction acting on a Hilbert space ${\displaystyle {\mathcal {H}}}$. We define some basic objects associated with T.

The defect operators of T are the operators D_T = (1 − T*T) and D_T* = (1 − TT*). The square root is the positive semidefinite one given by the spectral theorem. The defect spaces ${\displaystyle {\mathcal {D}}_{T}}$ and ${\displaystyle {\mathcal {D}}_{T*}}$ are the ranges Ran(D_T) and Ran(D_T*) respectively.

The defect indices of T are the pair

${\displaystyle ({\mbox{dim}}{\mathcal {D}}_{T},{\mbox{dim}}{\mathcal {D}}_{T^{*}}).}$

The defect operators and the defect indices are a measure of the non-unitarity of T.

A contraction T on a Hilbert space can be canonically decomposed into an orthogonal direct sum

${\displaystyle T=\Gamma \oplus U}$

where U is a unitary operator and Γ is completely non-unitary in the sense that it has no reducing subspaces on which its restriction is unitary. If U = 0, T is said to be a completely non-unitary contraction. A special case of this decomposition is the Wold decomposition for an isometry, where Γ is a proper isometry.

Contractions on Hilbert spaces can be viewed as the operator analogs of cos θ and are called operator angles in some contexts. The explicit description of contractions leads to (operator-)parametrizations of positive and unitary matrices.

Dilation theorem

Sz.-Nagy's dilation theorem, proved in 1953, states that for any contraction T on a Hilbert space H, there is a unitary operator U on a larger Hilbert space K ⊇ H such that if P is the orthogonal projection of K onto H then T =P U P for all n > 0. The operator U is called a dilation of T and is uniquely determined if U is mininal, i.e. K is the smallest closed subspace invariant under U and U* containing H.

In fact define

${\displaystyle \displaystyle {{\mathcal {H}}=H\oplus H\oplus H\oplus \cdots ,}}$

the orthogonal direct sum of countably many copies of H.

Let V be the isometry on ${\displaystyle {\mathcal {H}}}$ defined by

${\displaystyle \displaystyle {V(\xi _{1},\xi _{2},\xi _{3},\dots )=(T\xi _{1},{\sqrt {I-T^{*}T}}\xi _{1},\xi _{2},\xi _{3},\dots ).}}$

Let

${\displaystyle \displaystyle {{\mathcal {K}}={\mathcal {H}}\oplus {\mathcal {H}}.}}$

Define a unitary W on ${\displaystyle {\mathcal {K}}}$ by

${\displaystyle \displaystyle {W(x,y)=(Vx+(I-VV^{*})y,-V^{*}y).}}$

W is then a unitary dilation of T with H considered as the first component of ${\displaystyle {\mathcal {H}}\subset {\mathcal {K}}}$.

The minimal dilation U is obtained by taking the restriction of W to the closed subspace generated generated by powers of W applied to H.

Notes

1. Sz.-Nagy et al. 2010, p. 10-14

References

• Sz.-Nagy, B.; Foias, C.; Bercovici, H.; Kérchy, L. (2010), Harmonic analysis of operators on Hilbert space, Universitext (Second ed.), Springer, ISBN 978-1-4419-6093-1
• Riesz, F.; Sz.-Nagy, B. (1995), Functional analysis. Reprint of the 1955 original, Dover Books on Advanced Mathematics, Dover, pp. 466–472, ISBN 0-486-66289-6

• Operator theory