Unitary dilation

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 ${\displaystyle P_{H}}$ is an orthogonal projection on H.

V is said to be a unitary dilation (respectively, normal, isometric, etc.) if V is unitary (respectively, normal, isometric, etc.). T is said to be a compression of V. If an operator T has a spectral set ${\displaystyle X}$, we say that V is a normal boundary dilation or a normal ${\displaystyle \partial X}$ dilation if V is a normal dilation of T and ${\displaystyle \sigma (V)\in \partial X}$.

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