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In operator theory, a dilation of an operator T on a Hilbert space H is an operator on a larger Hilbert space K, whose restriction to H composed with the orthogonal projection onto H is T.
More formally, let T be a bounded operator on some Hilbert space H, and H be a subspace of a larger Hilbert space H'. A bounded operator V on H' is a dilation of T if
where P H {\displaystyle P_{H}} is an orthogonal projection on H.
V is said to be a unitary dilation if V is unitary. T is said to be a compression of V. If an operator T has a spectral set X {\displaystyle X} , we say that V is a normal boundary dilation or a normal ∂ X {\displaystyle \partial X} dilation if V is a normal dilation of T and σ ⊆ ∂ X {\displaystyle \sigma \subseteq \partial X}.