Schatten norm

Definition

Let H_1, H_2 be 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 = [\operatorname{Tr} (|T|^p)]^{1/p}, where |T|:=\sqrt{(T^*T)}, using the operator square root.

If T is compact and H_1,,H_2 are separable, then

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

for s_1(T) \ge s_2(T) \ge \cdots \ge s_n(T) \ge \cdots \ge 0 the singular values of T, i.e. the eigenvalues of the Hermitian operator |T|:=\sqrt{(T^*T)}.

Special cases

• The Schatten 1-norm is the nuclear norm (also known as the trace norm, or the Ky Fan n-norm).
• The Schatten 2-norm is the Frobenius norm.
• The Schatten ∞-norm is the spectral norm (also known as the operator norm, or the largest singular value).

Properties

In the following we formally extend the range of p to [1,\infty] with the convention that |\cdot|_{\infty} is the operator norm. The dual index to p=\infty is then q=1.

• The Schatten norms are unitarily invariant: for unitary operators U and V and p\in [1,\infty],

:: |U T V|_p = |T|_p.

• They satisfy Hölder's inequality: for all p\in [1,\infty] and q such that \frac{1}{p} + \frac{1}{q} = 1, and operators S\in\mathcal{L}(H_2,H_3), T\in\mathcal{L}(H_1,H_2) defined between Hilbert spaces H_1, H_2, and H_3 respectively, :: |ST|_1 \leq |S|_p |T|_q. If p,q,r\in [1,\infty] satisfy \tfrac{1}{p} + \tfrac{1}{q} = \tfrac{1}{r}, then we have :: |ST|_r \leq |S|_p |T|_q. The latter version of Hölder's inequality is proven in higher generality (for noncommutative L^p spaces instead of Schatten-p classes) in. (For matrices the latter result is found in.)

• Sub-multiplicativity: For all p\in [1,\infty] and operators S\in\mathcal{L}(H_2,H_3), T\in\mathcal{L}(H_1,H_2) defined between Hilbert spaces H_1, H_2, and H_3 respectively,

:: |ST|_p \leq |S|_p |T|_p .

• Monotonicity: For 1\leq p\leq p'\leq\infty,

:: |T|1 \geq |T|p \geq |T|{p'} \geq |T|\infty.

• Duality: Let H_1, H_2 be finite-dimensional Hilbert spaces, p\in [1,\infty] and q such that \frac{1}{p} + \frac{1}{q} = 1, then

:: |S|_p = \sup\lbrace |\langle S,T\rangle | \mid |T|_q = 1\rbrace,

: where \langle S,T\rangle = \operatorname{tr}(S^*T) denotes the Hilbert–Schmidt inner product.

• Let (e_k)k,(f{k'})_{k'} be two orthonormal basis of the Hilbert spaces H_1, H_2, then for p=1

:: |T|1 \leq \sum{k,k'}\left|T_{k,k'}\right|.

Remarks

Notice that |\cdot|_2 is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), |\cdot|1 is the trace class norm (see trace class), and |\cdot|\infty is the operator norm (see operator norm).

Note that the matrix p-norm is often also written as |\cdot|p, but it is not the same as Schatten norm. In fact, we have |A|{\text{matrix p-norm}, 2} = |A|_{\text{Schatten}, \infty}.

For p\in(0,1) the function |\cdot|_p is an example of a quasinorm.

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).

References

• Rajendra Bhatia, Matrix analysis, Vol. 169. Springer Science & Business Media, 1997.
• John Watrous, Theory of Quantum Information, 2.3 Norms of operators, lecture notes, University of Waterloo, 2011.
• Joachim Weidmann, Linear operators in Hilbert spaces, Vol. 20. Springer, New York, 1980.

References

1. Fan, Ky.. (1951). "Maximum properties and inequalities for the eigenvalues of completely continuous operators". Proceedings of the National Academy of Sciences of the United States of America.
2. (1986). "Generalized s-numbers of \tau-measurable operators.". Pacific Journal of Mathematics.
3. (1994). "Sharp uniform convexity and smoothness inequalities for trace norms". Inventiones Mathematicae.