LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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Finite-rank orthogonal compressions converge in trace norm

Statement

Assume Countable Choice. Let $H$ be a separable complex Hilbert space, let $T \in S_{1} (H)$ be trace class, and let $(P_{n})_{n \geq 1}$ be any supplied sequence of finite-rank orthogonal projections on $H$ such that $P_{n} \to I$ strongly. Then $∥ P_{n} T P_{n} - T ∥_{1} ⟶ 0.$ The projections need not be increasing. For example, the initial projections onto the first $n$ vectors of a supplied countable orthonormal basis satisfy the hypothesis.

Facts & Assumptions

Given: Countable Choice, separable complex $H$ , trace-class $T \in B (H)$ , and the specified finite-rank orthogonal projections $P_{n}$ .

[A1]

By the trace-class definition, a trace-class operator is compact (Trace class operator).

[A2]

Every finite-rank operator is trace class, and hence each SVD truncation $F_{N}$ is trace class (Trace class operator).

[A3]

Under Countable Choice the singular-value decomposition supplies orthonormal families $(e_{j})$ and $(f_{j})$ and the operator-norm convergent expansion $T x = \sum_{j} s_{j} ⟨ x, e_{j} ⟩ f_{j}$ ; its index set is finite exactly in the finite-rank case (Singular value decomposition for compact operators).

[A4]

If a trace-class operator $R$ has a nuclear representation $R = \sum_{j} ⟨ \cdot, u_{j} ⟩ v_{j}$ converging in operator norm, then $∥ R ∥_{1} \leq \sum_{j} ∥ u_{j} ∥ ∥ v_{j} ∥$ (Nuclear series characterizes trace norm).

[A5]

Trace-class operators form a linear space and the trace norm satisfies the triangle inequality (Trace class is a two sided Banach operator ideal).

[A6]

For bounded $A, B$ and trace-class $R$ , $∥ A R B ∥_{1} \leq ∥ A ∥ ∥ R ∥_{1} ∥ B ∥$ (Trace class is a two sided Banach operator ideal).

[A7]

Each orthogonal projection $P_{n}$ is self-adjoint (Hilbert projections are linear, self-adjoint and contractive).

[A8]

Each orthogonal projection satisfies $∥ P_{n} x ∥ \leq ∥ x ∥$ , hence $∥ P_{n} ∥ \leq 1$ (Hilbert projections are linear, self-adjoint and contractive).

[A9]

Strong convergence means pointwise norm convergence: $∥ P_{n} x - x ∥ \to 0$ for every fixed $x \in H$ (Strong and weak operator topologies).

[A10]

Countable Choice supplies the hypotheses of the trace-class, SVD, nuclear-series, ideal, orthogonal-projection, and Fourier-expansion results used below (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Its explicit countable basis-selection use here is through the SVD in [A3], which selects bases of its countably many finite-dimensional singular eigenspaces; no orthonormal basis of the ambient $H$ is separately chosen.

[A11]

The trace-class singular-value series converges, so its tails tend to zero (Trace class operator).

[A12]

The complex inner product is linear in its first argument, so for each positive real $s_{j}$ , $⟨ x, s_{j} e_{j} ⟩ = s_{j} ⟨ x, e_{j} ⟩$ (Real and complex inner-product spaces and their induced length).

[A13]

If $(g_{j})_{j \geq 1}$ is a supplied countable complete orthonormal family, then the initial Fourier sums $\sum_{j = 1}^{n} ⟨ x, g_{j} ⟩ g_{j}$ converge to each $x$ in norm (Orthonormal families, complete orthonormal systems and Hilbert bases, Fourier expansion in a Hilbert space).

[A14]

The orthogonal projection onto a closed subspace is characterized by $P_{M} x \in M$ and $x - P_{M} x \in M^{⊥}$ (The Hilbert orthogonal projection onto a closed subspace).

Proof

technique · direct

Given: The data in the statement and facts [A1]–[A14].

1.1A1A2A3A4A5A10A11

By [A1]–[A3] and [A10], write the SVD of $T$ and let $J$ be its positive-singular-value index set. For $N \geq 0$ set $F_{N} = \sum_{j \in J, j \leq N} s_{j} ⟨ \cdot, e_{j} ⟩ f_{j}$ , with $F_{0} = 0$ . Since $F_{N}$ is trace class by [A2], [A5] makes $R_{N} : = T - F_{N}$ trace class; the SVD gives the operator-norm convergent nuclear tail $R_{N} = \sum_{j \in J, j > N} ⟨ \cdot, s_{j} e_{j} ⟩ f_{j}$ , so [A4] gives $∥ R_{N} ∥_{1} \leq \sum_{j > N} s_{j} \to 0$ by [A11]. If $T$ has finite rank $r$ , then $R_{N} = 0$ for $N \geq r$ ; when $T = 0$ or $H = {0}$ the sum is empty and $F_{N} = R_{N} = 0$ .

1.2A10A12A13A14

For the stated basis example, let $(g_{j})_{j \geq 1}$ be the supplied complete orthonormal basis and let $P_{n}$ project onto $M_{n} : = span {g_{1}, \dots, g_{n}}$ . The sum $s_{n} : = \sum_{j = 1}^{n} ⟨ x, g_{j} ⟩ g_{j}$ lies in $M_{n}$ , and orthonormality plus [A12] gives $x - s_{n} \in M_{n}^{⊥}$ , so [A14] gives $s_{n} = P_{n} x$ ; now [A13] yields $P_{n} x \to x$ .

2.1A4A7A8A9A10A12step 1.1

Fix $N$ and the finite SVD sum $F_{N}$ from step 1.1; by [A12] write it as $F_{N} = \sum_{j \in J, j \leq N} ⟨ \cdot, u_{j} ⟩ v_{j}$ , where $u_{j} = s_{j} e_{j}$ and $v_{j} = f_{j}$ . Self-adjointness in [A7] gives $P_{n} F_{N} P_{n} = \sum_{j} ⟨ \cdot, P_{n} u_{j} ⟩ P_{n} v_{j}$ , so $P_{n} F_{N} P_{n} - F_{N} = \sum_{j} (⟨ \cdot, P_{n} u_{j} ⟩ (P_{n} v_{j} - v_{j}) + ⟨ \cdot, P_{n} u_{j} - u_{j} ⟩ v_{j})$ . This finite nuclear representation and [A4] bound its trace norm by $\sum_{j \in J, j \leq N} (∥ P_{n} u_{j} ∥ ∥ P_{n} v_{j} - v_{j} ∥ + ∥ P_{n} u_{j} - u_{j} ∥ ∥ v_{j} ∥)$ , which tends to zero by [A8]–[A9] and finiteness of the sum. Thus $∥ P_{n} F_{N} P_{n} - F_{N} ∥_{1} \to 0$ for each fixed $N$ , including the empty sum when $N = 0$ or $T = 0$ .

3.1A5A6A8A10step 1.1step 2.1

For every $n, N$ , the residual $R_{N}$ from step 1.1 satisfies $∥ P_{n} R_{N} P_{n} ∥_{1} \leq ∥ P_{n} ∥^{2} ∥ R_{N} ∥_{1} \leq ∥ R_{N} ∥_{1}$ by [A6] and [A8]. Decomposing $P_{n} T P_{n} - T = P_{n} R_{N} P_{n} + (P_{n} F_{N} P_{n} - F_{N}) - R_{N}$ using step 2.1, and applying [A5]'s trace-norm triangle inequality, yields $∥ P_{n} T P_{n} - T ∥_{1} \leq 2 ∥ R_{N} ∥_{1} + ∥ P_{n} F_{N} P_{n} - F_{N} ∥_{1}$ . All terms are trace class by [A5]–[A6].

4.1A3A10step 1.1step 2.1step 3.1∎

Given $ε > 0$ , choose $N$ by step 1.1 so that $∥ R_{N} ∥_{1} < ε / 4$ . For this fixed $N$ , step 2.1 gives an index $n_{0}$ such that $∥ P_{n} F_{N} P_{n} - F_{N} ∥_{1} < ε / 2$ for every $n \geq n_{0}$ . Step 3.1 then gives $∥ P_{n} T P_{n} - T ∥_{1} < ε$ for every $n \geq n_{0}$ , proving the claim. Countable Choice is the stated hypothesis of the trace-class, nuclear-series, ideal, orthogonal-projection, Fourier-expansion, and SVD suppliers in [A10]; the explicit countable basis selection used here is the SVD construction in [A3].

Depends on

Used by

Trace-norm continuity, growth and multiplicativity of the local determinant Lemma

Dependency tree · two levels

65 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kostenko, Trace Ideals with Applications, §3.4 (standard reference, not scraped)
van Neerven, Functional Analysis, §14.5.a (standard reference, not scraped)
Dyatlov–Zworski, Mathematical Theory of Scattering Resonances, App. B §§B.5–B.6 (standard reference, not scraped)