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[Math.SP] 3 Mar 2003 Σ and Operator Tnadfuirtasomof Transform Fourier Standard (1.2) Eta Ucin I Ore’ Neso Oml.(Here Formula

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[Math.SP] 3 Mar 2003 Σ and Operator Tnadfuirtasomof Transform Fourier Standard (1.2) Eta Ucin I Ore’ Neso Oml.(Here Formula FUNCTIONAL CALCULUS FOR NON-COMMUTING OPERATORS WITH REAL SPECTRA VIA AN ITERATED CAUCHY FORMULA MATS ANDERSSON & JOHANNES SJOSTRAND¨ Abstract. We define a smooth functional calculus for a non- commuting tuple of (unbounded) operators Aj on a Banach space with real spectra and resolvents with temperate growth, by means of an iterated Cauchy formula. The construction is also extended to tuples of more general operators allowing smooth functional calculii. We also discuss the relation to the case with commuting operators. 1. Introduction There are many different approaches to functional calculus for one or several operators acting on a Banach space, a common idea being that in order to define f(P ) where P is some operator and f a function of some suitable class, we represent f(x) as a superposition of sim- pler functions ωα(x), for which ωα(P ) can be defined and then define f(P ) as the corresponding superposition of the operators ωα(P ). For instance, if P is a self-adjoint operator on a Hilbert space, we have 1 (1.1) f(P )= f(t)eitP dt, 2π Z corresponding to the representation ofb f as a superposition of expo- nential functions via Fourier’s inversion formula. (Here f denotes the standard Fourier transform of f. This approach has been developed by M. Taylor [23] and others.) Another example is when P bis a bounded operator and f is holomorphic in a neighborhood of the spectrum, σ(P ), of P . Then 1 arXiv:math/0303024v1 [math.SP] 3 Mar 2003 (1.2) f(P )= f(z)(z − P )−1dz 2πi Zγ where γ is closed contour around σ(P ). For problems of spectral asymptotics and scattering for partial differ- ential operators, the representation (1.1) often has led to the sharpest known results (see H¨ormander [14], Ivrii [15]), but the price to pay Date: October 22, 2018. Key words and phrases. functional calculus, spectrum, non-commuting operators. First author partially supported by the Swedish Research Council, Second author invited to G¨oteborg University and Chalmers. 1 2 MATSANDERSSON&JOHANNESSJOSTRAND¨ is that one has to get a good understanding of the associated unitary group for instance via the theory of Fourier integral operators or via propagation estimates. Often a formula like (1.2) is easier and more practical to use. (See for instance Agmon–Kannai [1], Seeley [19].) The advantage is that the resolvent (z − P )−1 can be treated with simple means (like the theory of pseudodifferential operators). If P is bounded, f(z) is defined with its derivatives on the spectrum of P and has an extension f to a neighborhood of the spectrum such that ∂f vanishes to infinite order on σ(P ), and if the resolvent only e blows up polynomially when z tends to the spectrum, then Dynkin [11] used thee Cauchy-Green formula 1 −1 f(w)= − (z − w) ∂zf(z)L(dz), L(dz)= d(Re z)d(Im z), π Z e e to define 1 −1 (1.3) f(P )= − (z − P ) ∂zf(z)L(dz), π Z e and he studied the corresponding functional calculus (also with other classes of functions f allowing for wilder resolvent behaviour). This work has been very influencial (see below). Unknowingly of [11], Helffer and the second author [13] used (1.3) as a practical device in the study of magnetic Schr¨odinger operators in the framework of unbounded non-selfadjoint operators P ; f is then ∞ R the standard almost holomorphic extension of f ∈ C0 ( ). (We refrain from reviewing here the history of almost holomorphic extensionse with roots in the work of H¨ormander, Nirenberg, Dynkin and others.) It was soon realized that (1.3) is of great practical usefulness for many prob- lems in spectral and scattering theory and in mathematical physics, because it is simple to manipulate without requiring holomorphy of the test-functions f. For instance, if P is an elliptic differential oper- ator and f belongs to a suitable class of functions, it is very easy to show that f(P ) is a pseudodifferential operator ([13], [9]), and other applications were obtained in cases where f does not necessarily have compact support (E.B. Davies [8], A. Jensen, S. Nakamura [16]). An- other application of (1.3) is in the area of trace formulae and effective Hamiltonians: For a given operator P : H → H, one sometimes in- troduces an auxiliary (so called Grushin-, or in more special situations Feschbach-) problem: (1.4) (P − z)u + R−u− = v, R+v = v+. Here the auxiliary operators R+ : H → C+, R− : C− → H should be chosen in such a way that the problem (1.4) has a unique solution H ∋ u = Ev + E+v+, C− ∋ u− = E−v + E−+v+. FUNCTIONAL CALCULUS FOR NON-COMMUTING OPERATORS ... 3 for all v ∈ H, v+ ∈ C+. Then it is well-known that the operator E−+ inherits many of the properties of P , and typically one looks for spaces C± which are ”smaller” in some sense, so that the study of E−+ may be easier than that of P . For trace formulae one can show under quite general assumptions that 1 ∂f −1 dE−+ (1.5) tr f(P )=tr (z)(E−+) (z)L(dz). π Z ∂ze dz which is very useful for instance when the spaces C± are of finite (and here equal) dimensions. The approach of Dynkin [11] has had a great influence on many later works devoted to general problems of functional calculus. In [20] J. Taylor introduced a notion of joint spectrum σ(P ) ⊂ Cn for several commuting bounded operators P1,...,Pm on a Banach space, defined in terms of the mapping properties of the operators. This spec- trum is in general strictly smaller than the joint spectrum one ob- tains by regarding Pj as elements in some Banach subalgebra ofL( B). In [21] he then constructed a general holomorphic functional calculus Ø(σ(P )) → L( B) and proved basic functorial properties. In simple cases, for instance if the function f is entire, one can use a simple multiple Cauchy formula to represent f(P ), but the general case is intricate, and Taylor’s first construction was based on quite abstract Cauchy–Weil formulas; later on in [22] he made the whole construction with cohomological methods. In [2] was given a construction based on a multivariable notion of resolvent ωz−P which permits a representa- tion of the calculus analogous to formula (1.2). In special cases, for instance when the spectrum is real, such a representation was known earlier, and was used by Droste [10], following Dynkin’s approach (1.3), to obtain a smooth functional calculus in the multivariable case for op- erators with real spectra. This approach is extended to more general spectra in [18]. Various versions of functional calculus have been used in the study of the joint spectrum of several commuting selfadjoint operators ([7], [5, 6]), and for nonselfadjoint operators with real spectra in [4]. The case of non-commuting operators is more difficult and more challenging. The monograph of Nazaikinski, Shatalov and Sternin [17] gives a nice treatment of such a theory and contains references to many earlier works of V.P. Maslov and others. The authors build the the- ory on the approximation of functions of several variables by linear m combinations of tensor products. If f(x1, x2, ..., xm) = 1 fj(xj) is such a tensor product and Pj are operators on the same BanachQ space, that do not necessarily commute, it is natural to define f(P1, ..., Pm) as f1(P1) ◦ ... ◦ fm(Pm), and then approximate a general f(x1, ..., xm) by linear combinations of tensor products, and define f(P1, ..., Pm) as 4 MATSANDERSSON&JOHANNESSJOSTRAND¨ the corresponding limit in the space of operators. A prototype for non-commutative functional calculus is given by the theory of pseudo- differential operators, with x1, x2, ..., xn,Dx1 , ..., Dxn as the basic set of non-commuting operators. Most approaches to the theory of pseudodifferential operators use direct methods rather than approximation by tensor products. In this paper we shall suggest a direct approach to smooth non-commutative functional calculus, based on a multivariable version (3.3) of (1.3). (An- other possibility, that will not be explored here is to extend (1.1) to the multivariable case. Then, under suitable extra assumptions, one could also consider the Weyl quantization w −1 it·P f (P1,...,Pm)=(2π) f(t)e dt, Z b with t·P = tjPj.) When P1,...,Pm are pseudodifferential operators with real principalP symbols and f belongs to a suitable symbol class, it will be quite obvious from our formula that f(P1,...,Pm) is also a pseu- dodifferential operator, by extending the arguments from [13], [9]. We hope that the multivariable formula (3.3) will be a useful complement to existing multivariable functional calculii. It might provide a more direct alternative to some parts of the theory of in [17]. The purpose of the present paper is merely to establish some basis for this approach and to connect it to the one of J. Taylor and others ([20, 21, 4, 2, 3]) in the commutative case. The plan of the paper is the following: In Section 2, we introduce some special almost holomorphic exten- sions of smooth functions on the real domain. In Section 3 we introduce the calculus using the formula (3.3). and in Section 4 we establish some additional properties. Thus we get a ∞ C0 -calculus of several unbounded and non-commuting operators whose spectra are real and which have locally temperate growth of the resol- vent near the real axis.
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