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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We prove a global limiting absorption principle on the entire real line for free, massless Dirac operators H 0 = α · (−i∇) for all space dimensions n ∈ N, n 2. This is a new result for all dimensions other than three, in particular, it applies to the two-dimensional case which is known to be of some relevance in applications to graphene.We also prove an essential self-adjointness result for first-order matrixvalued differential operators with Lipschitz coefficients.

Abstract. Using the general formalism of [12], a study of index theory for non-Fredholm operators was initiated in [9]. Natural examples arise from (1 + 1)-dimensional differential operators using the model operator, and the family of self-adjoint operators A(t) in L 2 (R; dx) studied here is explicitly given byHere φ : R → R has to be integrable on R and θ : R → R tends to zero as t → −∞ and to 1 as t → +∞ (both functions are subject to additional hypotheses). In particular, A(t), t ∈ R, has asymptotes (in the norm resolvent sense)The interesting feature is that D A violates the relative trace class condition introduced in [9, Hypothesis 2.1 (iv)]. A new approach adapted to differential operators of this kind is given here using an approximation technique. The approximants do fit the framework of [9] enabling the following results to be obtained. Introducingwhenever this limit exists. In the concrete example at hand, we prove2πˆR dx φ(x). Here ξ( · ; S 2 , S 1 ) denotes the spectral shift operator for the pair of self-adjoint operators (S 2 , S 1 ), and we employ the normalization, ξ(λ; H 2 , H 1 ) = 0, λ < 0. Contents

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