Schatten norm

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In mathematics, specifically functional analysis, the Schatten norm (or Schatten–von-Neumann norm) arises as a generalization of p-integrability similar to the trace class norm and the Hilbert–Schmidt norm.

Definition[edit]

Let H_1, H_2 be separable Hilbert spaces, and T a (linear) bounded operator from H_1 to H_2. For p\in [1,\infty), define the Schatten p-norm of T as

 \|T\| _{p} := \bigg( \sum _{n\ge 1} s^p_n(T)\bigg)^{1/p}

for  s_1(T) \ge s_2(T) \ge \cdots s_n(T) \ge \cdots \ge 0 the singular values of T, i.e. the eigenvalues of the Hermitian matrix |T|:=\sqrt{(T^*T)}. From functional calculus on the positive operator T * T it follows that

 \|T\| _{p}^p = \mathrm{tr} (|T|^p)


Remarks[edit]

The Schatten norm is unitarily invariant: for  U and  V unitary operators,

 \|U T V\|_p = \|T\|_p

Notice that \|\ \|_2 is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator) and \|\ \|_1 is the trace class norm (see trace class).

An operator which has a finite Schatten norm is called a Schatten class operator and the space of such operators is denoted by  S_p(H_1,H_2). With this norm,  S_p(H_1,H_2) is a Banach space, and a Hilbert space for p = 2.

Observe that  S_p(H_1,H_2) \subseteq \mathcal{K} (H_1,H_2), the algebra of compact operators. 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).