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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 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 , we say that v is a normal boundary dilation or a normal dilation if v is a normal dilation of t and . some texts impose an additional condition. namely, that a dilation satisfy the following (calculus) property: : where f(t) is some specified functional calculus (for example, the polynomial or h∞ calculus). the utility of a dilation is that it allows the "lifting" of objects associated to t to the level of v, where the lifted objects may have nicer properties. see, for example, the commutant lifting theorem.