Unitary operators | Operator theory
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. (Wikipedia).
👉 Learn about dilations. Dilation is the transformation of a shape by a scale factor to produce an image that is similar to the original shape but is different in size from the original shape. A dilation that creates a larger image is called an enlargement or a stretch while a dilation tha
From playlist Transformations
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From playlist DTU: Mathematics 4 Real Analysis | CosmoLearning.org Math
👉 Learn about dilations. Dilation is the transformation of a shape by a scale factor to produce an image that is similar to the original shape but is different in size from the original shape. A dilation that creates a larger image is called an enlargement or a stretch while a dilation tha
From playlist Transformations
What is an enlargement dilation
👉 Learn about dilations. Dilation is the transformation of a shape by a scale factor to produce an image that is similar to the original shape but is different in size from the original shape. A dilation that creates a larger image is called an enlargement or a stretch while a dilation tha
From playlist Transformations
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How to determine the scale factor of a dilation for enlargement
👉 Learn about dilations. Dilation is the transformation of a shape by a scale factor to produce an image that is similar to the original shape but is different in size from the original shape. A dilation that creates a larger image is called an enlargement or a stretch while a dilation tha
From playlist Transformations
How to determine the ratio of a reduction dilation
👉 Learn about dilations. Dilation is the transformation of a shape by a scale factor to produce an image that is similar to the original shape but is different in size from the original shape. A dilation that creates a larger image is called an enlargement or a stretch while a dilation tha
From playlist Transformations
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From playlist Mathematics
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What Are Dilutions | Chemical Calculations | Chemistry | FuseSchool
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