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In this paper, we extend and strengthen classical estimates for singular values of integral operators originally due to Cwikel, Birman and Solomyak. Suppose that A1 and A2 are two semifinite von Neumann algebras with representations π1:scriptA1→Bfalse(scriptHfalse) and π2:scriptA2→Bfalse(scriptHfalse) such that false∥π1false(xfalse)π2false(yfalse)∥scriptL2false(scriptHfalse)⩽const ∥x∥scriptL2false(A1false)∥y∥scriptL2false(scriptAfalse).Our first result asserts that the inequality false∥π1false(xfalse)π2false(yfalse)∥E(H)⩽CE∥x⊗y∥E(scriptA1⊗scriptA2)holds in any interpolation space E for (L2,L∞). In the special case, when scriptA1=scriptA2=L∞false(Rdfalse) and π1, π2 are given by multipliers and Fourier multipliers, respectively, our result yields a (strengthened version) of well‐known Cwikel estimates [Cwikel, Ann. of Math. (2) 106 (1977) 93–100]. We demonstrate further the applicability of our result by considering noncommutative Euclidean space (Moyal plane) and magnetic Laplacian. Our second direction relates to Birman–Solomyak estimates for interpolation space E for the (quasi)‐Banach couple (ℓp,ℓ2), p<2. In this setting, our technique yields substantial strengthening of results from [Birman and Solomyak, Russian Math. Surveys 32 (1977) 15–89] and Chapter VI of Simon [Trace ideals and their applications (American Mathematical Society, Providence, RI, 2005)]. Finally, we provide Cwikel–Birman–Solomyak estimates for the crucial case of weak Schatten p‐ideals, 1⩽p<2, in the setting of noncommutative Euclidean space.

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Let M be a semi-finite von Neumann algebra and let f :. This result resolves a conjecture raised by F. Nazarov and V. Peller implying a couple of existing results in perturbation theory.

We introduce a new approach to traces on the principal ideal L 1,∞ generated by any positive compact operator whose singular value sequence is the harmonic sequence. Distinct from the well-known construction of J. Dixmier, the new approach provides the explicit construction of every trace of every operator in L 1,∞ in terms of translation invariant functionals applied to a sequence of restricted sums of eigenvalues. The approach is based on a remarkable bijection between the set of all traces on L 1,∞ and the set of all translation invariant functionals on l ∞ . This bijection allows us to identify all known and commonly used subsets of traces (Dixmier traces, Connes-Dixmier traces, etc.) in terms of invariance properties of linear functionals on l ∞ , and definitively classify the measurability of operators in L 1,∞ in terms of qualified convergence of sums of eigenvalues. This classification has led us to a resolution of several open problems (for the class L 1,∞ ) from [7]. As an application we extend Connes' classical trace theorem to positive normalised traces.