Schatten p classes

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H$ and $K$ be real or complex Hilbert spaces and let $T\in \mathcal{B}(H,K)$ be compact (Compact linear operator) with zero-padded singular-value sequence $(s_n(T){)}_{n\ge 1}$ (Absolute value and singular values of a compact operator). For a real $1\le p<+\mathrm{\infty }$ the Schatten $p$-class ${\mathcal{S}}_p(H,K)$ is defined to be the set of compact operators with $\mathrm{\parallel }T{\mathrm{\parallel }}_p:=({\sum }_{n\ge 1}{s}_n(T{)}^p{)}^{1\mathrm{/}p}<+\mathrm{\infty }\mathrm{.}$ This is an orientation remark only. It records the scale of ideals without developing any of its theory.

The two endpoints that this page does develop are recognised as follows. ${\mathcal{S}}_1(H,K)$ is exactly the trace-class ideal and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1$ is the trace norm, by definition of the latter (Trace class operator), including the zero padding and the finite-rank case. ${\mathcal{S}}_2(H,K)$ is exactly the class of operators that are Hilbert–Schmidt relative to a supplied Hilbert basis $E$ of $H$: for such a basis, let $(e_j{)}_{j\in J}\subseteq H$ and $(f_j{)}_{j\in J}\subseteq K$ be the right and left singular families supplied by the SVD. Its expansion gives $Te={\sum }_{j\in J}{s}_j⟨e,e_j⟩f_j,$ so orthonormality of the $f_j$, Parseval for the supplied basis $E$, and the interchange of the nonnegative finite-subset suprema give ${\sum }_{e\in E}\mathrm{\parallel }Te{\mathrm{\parallel }}^2={\sum }_{n\ge 1}{s}_n(T{)}^2,$ since ${\sum }_{e\in E}\mathrm{\parallel }Te{\mathrm{\parallel }}^2={\sum }_{j\in J}{s}_j^2{\sum }_{e\in E}\mathrm{\mid }⟨e,e_j⟩{\mathrm{\mid }}^2={\sum }_{j\in J}{s}_j^2,$ and the last sum is the zero-padded singular-value sum (Singular value decomposition for compact operators, Hilbert–Schmidt operator and Hilbert–Schmidt norm, Parseval equivalences for an orthonormal family, Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$, Orthonormal families, complete orthonormal systems and Hilbert bases), so that, when the common sum is finite, $\mathrm{\parallel }T{\mathrm{\parallel }}_2=\mathrm{\parallel }T{\mathrm{\parallel }}_{HS,E}$. For clarity, the interchange uses only finite rectangles: every finite subset of $E\times J$ lies in the product of its two finite projections, and finitely many finite sets of indices have finite union. Thus both iterated nonnegative suprema equal the supremum over finite rectangles, even when they are infinite. The norm identity for each $Te$ follows first for finite orthogonal sums and then by norm convergence of its SVD expansion. Parseval is applied in $H$ to $e_j$, with $\mathrm{\mid }⟨e,e_j⟩\mathrm{\mid }=\mathrm{\mid }⟨e_j,e⟩\mathrm{\mid }$. Conversely, every bounded operator Hilbert–Schmidt relative to $E$ is compact by Hilbert–Schmidt operators are compact, so the same calculation applies to it and puts it in ${\mathcal{S}}_2(H,K)$. No basis of $K$ is required, and the norm retains the notation $\mathrm{\parallel }T{\mathrm{\parallel }}_{HS,E}$ of its definition. The formula proves the same value for every supplied basis of $H$ in this compact-operator setting; it does not assert existence of such a basis. Empty singular families and an empty domain basis contribute zero. For $p=+\mathrm{\infty }$ one writes ${\mathcal{S}}_{\mathrm{\infty }}(H,K)$ for the compact operators with the operator norm $\mathrm{\parallel }T{\mathrm{\parallel }}_{S_{\mathrm{\infty }}}:=\mathrm{\parallel }T\mathrm{\parallel }=s_1(T)$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

Deliberate boundaries. Nothing here asserts completeness of $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_p$, Hölder or Young inequalities, duality, interpolation, or the identification of ${\mathcal{S}}_p$ with a space of sequences; no monotonicity of the norms beyond $\mathrm{\parallel }T\mathrm{\parallel }\le \mathrm{\parallel }T{\mathrm{\parallel }}_1$ from the trace-class page is claimed. Later items must not use this remark as a supplier: it is recorded for orientation, exactly as the functional-analysis plan's FA-16 boundary requires, and the general theory of Schatten classes belongs to a later, separate development.