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| In [[mathematics]], specifically [[functional analysis]], the '''[[Robert Schatten|Schatten]] norm''' (or '''Schatten- Von-Neumann norm''')
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| arises as a generalization of ''p''-integrability similar to the [[trace class]] [[norm (mathematics)|norm]] and the [[Hilbert–Schmidt operator|Hilbert–Schmidt]] norm.
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| ==Definition==
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| Let <math>H_1</math>, <math>H_2</math> be separable Hilbert spaces, and <math>T</math> a (linear) bounded operator from
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| <math>H_1</math> to <math>H_2</math>. For <math>p\in [1,\infty)</math>, define the Schatten p-norm of <math>T</math> as
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| : <math> \|T\| _{p} := \bigg( \sum _{n\ge 1} s^p_n(T)\bigg)^{1/p} </math> | |
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| for <math> s_1(T) \ge s_2(T) \ge \cdots s_n(T) \ge \cdots \ge 0</math>
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| the [[singular values]] of <math>T</math>, i.e. the eigenvalues of the Hermitian matrix <math>|T|:=\sqrt{(T^*T)}</math>.
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| From [[functional calculus]] on the positive operator ''T*T'' it follows that
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| : <math> \|T\| _{p}^p = \mathrm{tr} (|T|^p) </math>
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| ==Remarks==
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| The Schatten norm is unitarily invariant: for <math> U </math> and <math> V </math> unitary operators,
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| : <math> \|U T V\| _{p} = \|T\| _{p} </math>
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| Notice that <math>\|\ \| _{2} </math> is the Hilbert-Schmidt norm (see [[Hilbert-Schmidt operator]]) and
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| <math>\|\ \| _{1} </math> is the trace class norm
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| (see [[trace class]]).
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| An operator which has a finite Schatten norm is called a [[Schatten class operator]] and the space of such operators is denoted by <math> S_p(H_1,H_2)</math>. With this norm, <math> S_p(H_1,H_2)</math> is a Banach space, and a Hilbert space for ''p=2''.
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| Observe that <math> S_p(H_1,H_2) \subseteq \mathcal{K} (H_1,H_2)</math>, the algebra of [[compact operator]]s. This follows from the fact that if the sum is finite the spectrum will be finite or countable with the origin as limit point, and hence a compact operator (see [[compact operator on Hilbert space]]).
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| [[Category:Operator theory]]
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Materials Engineer Reinhard from Don Mills, has pastimes which include geocaching, injustice gods among us and hockey. Gains lots of motivation from life by visiting spots like Palace and Park of Fontainebleau.
my web site :: injustice gods among us wiki superman