The Spectral Theory of Toeplitz Operators. (AM-99)

Monvel, L. Boutet de; Guillemin, Victor

doi:10.1515/9781400881444

Cited by 322 publications

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“…Once one has chosen a global phase function, it is obviously always possible to choose an elliptic symbol -of any prescribed degree (cf. also [5]) 1 . Note that elliptic only means that the top symbol is invertible on the real part of the canonical relation, not that the operator is invertible mod.…”

Section: Definition 1 a Fourier Integral Operator A Is Adapted To U Imentioning

confidence: 93%

“…The Toeplitz operators and Toeplitz algebras used here, associated to a CR or a contact structure, were introduced and studied in [4,5], using the analysis of the singularity of the Szegö kernel (cf. [3,19]), or in a weaker form, the "Hermite calculus" of [2,11].…”

Section: Adapted Fourier Integral Operatorsmentioning

confidence: 99%

“…Adapted FIO always exist (cf [5]), more precisely Proposition 1 For any symplectic isomorphism u as above, there exists an elliptic FIO adapted to u.…”

Section: Definition 1 a Fourier Integral Operator A Is Adapted To U Imentioning

confidence: 99%

“…This is mentioned in [5]; the crux of the matter is that if Q(y) is a quadratic form with ≫ 0 real part, the integral e −|ξ|Q(y) dy does not vanish: it is an elliptic symbol of degree − 1 2 ny, equal to disc(…”

Section: Its Canonical Relation Is Transversally Defined and Positivementioning

confidence: 99%

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Vanishing of the logarithmic trace of generalized Szegö projectors

Monvel

2008

Algebraic Analysis of Differential Equations

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we show that the logarithmic trace of Szegö projectors introduced by K. Hirachi [15] for CR structures and extended in [8] to contact structures vanishes identically.Keywords: CR manifolds, contact manifolds, Toeplitz operators, residual trace. MSC2000: 58J40, 32A25, 53D10, 53D55.In [15] K. Hirachi showed that the logarithmic trace of the Szegö projector is an invariant of the CR structure. In [8] I showed that it is also defined for generalized Szegö projectors associated to a contact structure (definitions recalled below, sect.4), that it is a contact invariant, and that it vanishes if the base manifold is a 3-sphere, with arbitrary contact structure (not necessarily the canonical one). Here we show that it always vanishes. For this use the fact that this logarithmic trace is the residual trace of the identity (definitions recalled below, sect.5), and show that it always vanishes, because the Toeplitz algebra associated to a contact structure can be embedded in the Toeplitz algebra of a sphere, where the identity maps of all 'good' Toeplitz modules have zero residual trace. NotationsWe first recall the notions that we will use. Most of the material below in §1-5 is not new; we have just recalled briefly the definitions and useful properties, and send back to the literature for further details (cf. [16,17,21,18]).If X is a smooth manifold we denote T • X ⊂ T * X the set of non-zero covectors. A complex subspace Z corresponds to an ideal IZ ⊂ C ∞ (T • X, C)). Z is conic (homogeneous) if it is generated by homogeneous functions. It is smooth if IZ is locally generated by k = codim Z functions with linearly independent derivatives. If Z is smooth, it is involutive if IZ is stable by the Poisson bracket (in local coordinates {f, g} = ∂f ∂ξ j ∂g ∂x j − ∂f ∂x j ∂g ∂ξ j ); it is ≫ 0 if locally IZ has generators ui, vj (1 ≤ i ≤ p, 1 ≤ j ≤ q, p + q =codim Z) such that the vj are real, ui complex, and the matrix 1 i ({u k ,ū l ) is hermitian ≫ 0. The real part ZR is then a smooth real submanifold of T • X, whose ideal is generated by the Re ui, Im ui, vj. If I is ≫ 0, it is exactly determined by its formal germ (Taylor expansion) along the set of real points of ZR.A Fourier integral operator (FIO) from Y to X, is a linear operator from functions or distributions on Y to same on X defined as a locally finite sum of oscillating integralswhere φ is a phase function (homogeneous w.r. to θ), a a symbol function. Here we will only consider regular symbols, i.e. which are asymptotic sums a ∼ k≥0 a m−k where a m−k is homogeneous of degree m − k, k an integer. m could be any complex number. There is also a notion of vector FIO, acting on sections of vector bundles, which we will not use here.

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Section: Definition 1 a Fourier Integral Operator A Is Adapted To U Imentioning

confidence: 93%

Section: Adapted Fourier Integral Operatorsmentioning

confidence: 99%

“…Adapted FIO always exist (cf [5]), more precisely Proposition 1 For any symplectic isomorphism u as above, there exists an elliptic FIO adapted to u.…”

Section: Definition 1 a Fourier Integral Operator A Is Adapted To U Imentioning

confidence: 99%

Section: Its Canonical Relation Is Transversally Defined and Positivementioning

confidence: 99%

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Vanishing of the logarithmic trace of generalized Szegö projectors

Monvel

2008

Algebraic Analysis of Differential Equations

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“…A remarkable connection between the theory of Toeplitz operatorsá la [1] and the standard theory of pseudodifferential operators emerged from the work [5]. This connection in its broad outlines is elucidated in [4], too.…”

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confidence: 98%

A Class of Toeplitz Operators in Several Complex Variables

Fedchenko¹

2017

J. Sib. Fed. Univ., Math. phys.

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In order to study the Toeplitz algebras related to a Dirac operators in a neighborhood of a closed bounded domain D with smooth boundary in C n we introduce a singular Cauchy type integral. We compute its principal symbol, thus initiating the index theory.Keywords: Dirac operators, Cauchy type integral, symbol, Toeplitz operators, index. DOI: 10.17516/1997-1397-2017 There are a number of ways in which the theory of Toeplitz operators can be generalised to n dimensions, see e.g. This work focuses on a new class of Toeplitz operators which is more closely related to elliptic theory. The new Toeplitz operators admit very transparent description which motivates strikingly their study. To this end, let A be a (k×k) -matrix of first order scalar partial differential operators in a neighborhood of the closed bounded domain D with smooth boundary S in C n . Assume that the leading symbol of A has rank k away from the zero section of the cotangent bundle of D. Then, given any solution u of the homogeneous equation Au = 0 in the interior of D which has finite order of growth at the boundary, the Cauchy data t(u) of u with respect to A possess weak limit values at the boundary. If A satisfies the so-called uniqueness condition of the local Cauchy problem in a neighborhood of D, then the solution u is uniquely defined by its Cauchy data at S. Let t(u) = Bu be a representation of the Cauchy data of u. The space of all Cauchy data Bu of u at the boundary is effectively described by the condition of orthogonality to solutions of the formal adjoint equation A * g = 0 near D by means of a Green formula, see [6, § 10.3.4]. In this way we distinguish Hilbert space of vector-valued functions on S which represent solutions to Au = 0 in the interior of D. In particular, one introduces Hardy spaces H as subspaces of L 2 (S, C k ) consisting of the Cauchy data of solutions to Au = 0 in the interior of D with appropriate behaviour at the boundary. Pick such a Hilbert space H. By the above, H is a closed subspace of L 2 (S, C k ) and we write Π for the orthogonal projection of L 2 (S, C k ) onto H. If A is the Cauchy-Riemann operator then Π just amounts to the Szegö projection.Given a (k × k) -matrix M (z) of smooth function on S, the operator T M on H given by u → Π(M u) is said to be a Toeplitz operator with multiplier M . If M is a scalar multiple of the * fdp@bk.ru c ⃝ Siberian Federal University. All rights reserved -206 -

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Viscosity Limits for Zeroth‐Order Pseudodifferential Operators

Galkowski

Zworski

2022

Comm Pure Appl Math

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Motivated by the work of Colin de Verdière and Saint-Raymond on spectral theory for zeroth-order pseudodifferential operators on tori, we consider viscosity limits in which zeroth-order operators, P , are replaced by P C i , > 0. By adapting the Helffer-Sjöstrand theory of scattering resonances, we show that, in a complex neighbourhood of the continuous spectrum, eigenvalues of P C i have limits as the viscosity goes to 0. In the simplified setting of tori, this justifies claims made in the physics literature.

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The Spectral Theory of Toeplitz Operators. (AM-99)

Cited by 322 publications

References 0 publications

Vanishing of the logarithmic trace of generalized Szegö projectors

Vanishing of the logarithmic trace of generalized Szegö projectors

A Class of Toeplitz Operators in Several Complex Variables

Viscosity Limits for Zeroth‐Order Pseudodifferential Operators

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