Contraction (operator theory)

In operator theory, a discipline within mathematics, 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. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias.

If T is a contraction acting on a Hilbert space ${\displaystyle {\mathcal {H}}}$, the following basic objects associated with T can be defined.

The defect operators of T are the operators D_T = (1 − T*T)^½ and D_T* = (1 − TT*)^½. 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 positive operator D_T induces an inner product on ${\displaystyle {\mathcal {H}}}$. The inner product space can be identified naturally with Ran(D_T). A similar statement holds for ${\displaystyle {\mathcal {D}}_{T*}}$.

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