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Fredholm Theory of Toeplitz Operators on Doubling Fock Hilbert Spaces

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Fredholm Theory of Toeplitz Operators on Doubling Fock Hilbert Spaces Fredholm theory of Toeplitz operators on doubling Fock Hilbert spaces Article Accepted Version Al-Qabani, A., Hilberdink, T. and Virtanen, J. A. (2020) Fredholm theory of Toeplitz operators on doubling Fock Hilbert spaces. Mathematica Scandinavica, 126 (3). pp. 593-602. ISSN 1903-1807 doi: https://doi.org/10.7146/math.scand.a- 120920 Available at http://centaur.reading.ac.uk/85955/ It is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing . Published version at: https://www.mscand.dk/forthcoming To link to this article DOI: http://dx.doi.org/10.7146/math.scand.a-120920 Publisher: Mathematica Scandinavica All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs online FREDHOLM THEORY OF TOEPLITZ OPERATORS ON DOUBLING FOCK HILBERT SPACES AAMENA AL-QABANI, TITUS HILBERDINK AND JANI A. VIRTANEN Abstract. We study the Fredholm properties of Toeplitz operators acting on doubling Fock Hilbert spaces, and describe their essential spectra for bounded symbols of vanishing oscillation. We also compute the index of these Toeplitz operators in the special case when '(z) = jzjβ with β > 0. Our work extends the recent results on Toeplitz operators on the standard weighted Fock spaces to the setting of doubling Fock spaces. 1. Introduction A positive Borel measure µ on the complex plane C is said to be a doubling measure if there is a positive constant C such that µ (D(z; 2r)) ≤ Cµ (D(z; r)) ; for all z 2 C and r > 0, where D(z; r) = fw 2 C : jw − zj < rg: We denote by dA the standard Lebesgue area measure on C. Let ' be a subharmonic non- harmonic real-valued function of class C2 on the complex plane C such that ∆' dA is a doubling measure, where ∆' is the Laplacian of the function ' defined by ∆' = 'xx + 'yy: 2 The doubling Fock space F' is defined by Z 2 2 2 −2'(z) F' = f 2 H(C): kfk' = jf(z)j e dA(z) < 1 ; C where H(C) is the set of all entire functions. These spaces were introduced in [6], where their sampling and interpolating sequences were described. 2 It is well known that F' is a Hilbert space with inner product given by Z hf; gi = f(w)g(w)e−2'(w)dA(w): C We also note that the doubling Fock spaces include the standard weighted Fock spaces [14], the Fock-Sobolev space [3], the Fock spaces with weights '(z) = jzjβ, where β > 0 (see [10]), and the generalized Fock spaces [11] with weights ' satisfying 0 < C1 ≤ ∆'(z) ≤ C2, for all z 2 C, where C1 and C2 are positive constants. 2010 Mathematics Subject Classification. Primary 47B35; Secondary 30H20. Key words and phrases. Toeplitz operators, Hankel operators, doubling Fock spaces, Fredholm properties, essential spectrum. J. Virtanen was supported in part by Engineering and Physical Sciences Research Council grant EP/M024784/1. 1 2 AL-QABANI, HILBERDINK AND VIRTANEN 2 When ∆' dA is doubling, we say that a measurable function f : C ! C is in L' if Z 2 2 −2'(z) kfk' = jf(z)j e dA(z) < 1: C 2 2 We denote by P the orthogonal projection of L' onto F'. It has the integral representation Z −2'(w) P f(z) = f(w)Kz(w)e dA(w); C where Kz is the reproducing kernel function; that is, for each z 2 C, Kz is the unique 2 function in F' for which f(z) = hf; Kzi; 2 for all f 2 F'. Note that Kz depends on ' and we write Kz(w) = K'(w; z) = K(w; z); for z; w 2 C. For z 2 C, the normalized reproducing kernel function at z is defined by K'(w; z) kz(w) = ; kK'(z; z)k' for w 2 C. 1 1 2 For f 2 L = L (C), the Toeplitz operator Tf on F' with symbol f is defined by Tf (g) = P (fg): 2 It is clearly a bounded linear operator on F' and kTf k ≤ kfk1. We say that a bounded linear operator T on a Banach space X is Fredholm if ker T and X=T (X) are both finite-dimensional. In this case, the index of T is defined to be ind T = dim ker T − dim (X=T (X)) : The essential spectrum σess(T ) is defined by σess(T ) = fλ 2 C : T − λI is not Fredholmg; where I denotes the identity operator on X. Clearly σess(T ) is contained in the spectrum σ(T ) of T . Our main results are concerned with the Fredholm properties and computation of the 2 index of Toeplitz operators acting on doubling Fock spaces F' with bounded symbols of vanishing oscillation. Before introducing the notion of vanishing oscillation, we make a few useful observations. Since dµ = ∆' dA is a doubling measure, by [12, xI.8.6], for any z 2 C and r > 0, we have that µ (@(D(z; r)) = µ(fzg) = 0 and µ (D(z; 2r)) ≥ Cµ (D(z; r)), where C > 1 is a constant. Moreover, our assumption that ' is non-harmonic implies that µ is a locally finite non-zero doubling measure on C, so 0 < µ (D(z; r)) < 1, for any z 2 C and r > 0. It follows that, for each z 2 C, µ (D(z; r)) ! 1 as r ! 1, and the function r 7! µ (D(z; r)) is an increasing homeomorphism from the interval (0; 1) onto itself. Hence, for every z 2 C, there is a unique radius ρ(z) such that µ(D(z; ρ(z)) = 1: TOEPLITZ OPERATORS ON DOUBLING FOCK SPACES 3 For z 2 C and r > 0, we denote by Dr(z) the disk D(z; rρ(z)). Then we say that f is of vanishing oscillation and write f 2 VO if f is a continuous function on C and !r(f)(z) = sup jf(z) − f(w)j ! 0 w2Dr(z) as z ! 1. Note that VO is independent of the choice of r. 2. Main results The Fredholm properties of Toeplitz operators on the unweighted Fock space F 2 were described by Berger and Coburn [2] in 1987 using heavy machinery of C∗-algebras suitable for operators on Hilbert spaces. In 1992, Stroethoff [13] provided elementary proofs of their p results, and very recently the theory was extended to the standard weighted Fock spaces Fα using elementary methods [1] and limit operator techniques [4]. In the following theorem, we 2 extend the theory to doubling Fock Hilbert spaces F', which also sets the stage for further p extensions to other doubling Fock spaces F'. 1 2 Theorem 1. Let f 2 L \ VO. Then the Toeplitz operator Tf on F' is Fredholm if and only if there is a radius R > 0 such that f is bounded away from zero on C nD(0;R), that is, infjz|≥R jf(z)j > 0. In this case, \ σess(Tf ) = cl f (C nD(0;R)) ; R>0 where cl stands for the closure of the given set. 2 The difficulty with computing the index of Toeplitz operators on F' is that doubling Fock spaces do not have simple bases unlike the standard weighted Fock spaces where index formulas can be obtained more easily; see [2] for Toeplitz operators on F 2 and [1] for these p 1 β operators on Fα. The following result gives an index formula in the special case '(z) = 2 jzj with β > 0, which was introduced in [9] to study the properties of Hankel operators. For 2 2 2 convenience, we write Fjzjβ for F 1 β . It is not difficult to see that Fjzjβ is a doubling Fock 2 jzj space (and not a standard weighted Fock space). Theorem 2. Let f 2 L1 \ VO. Then the Toeplitz operator T on F 2 is Fredholm if and f jzjβ only if there is a radius R > 0 such that f is bounded away from zero on C nD(0;R). In this case, ind(Tf ) = − wind(fjfjzj=Rg); where wind(fjfjzj=Rg) is the winding number of the curve f(fjzj = Rg) around the origin. The proofs of our main theorems are given in Section 4 below. 3. Preliminaries 2 The following estimate for the Bergman kernel for F' plays an important role in the study of concrete operators (such as Toeplitz and Hankel operators) on doubling Fock spaces (see [5, 8]). 4 AL-QABANI, HILBERDINK AND VIRTANEN Theorem 3 ([7, Theorem 1.1 and (3)]). There are constants C > 0 and > 0 such that 1 e'(z1)+'(z2) jK(z1; z2)j ≤ C ; ρ(z1)ρ(z2) exp (d'(z1; z2) ) −2 for any z1; z2 2 C, where d' is the distance on C induced by the metric '(z) dz ⊗ dz¯. The previous theorem combined with the estimate (see [7, Proposition 2.10]) e'(z) e'(z) C−1 ≤ kK (·; z)k ≤ C ; for z 2 ; (1) ρ(z) ' ' ρ(z) C where C is a positive constant, gives the following result. Lemma 4 ([8, Lemma 2.7]). Let r ≥ 1. For every k ≥ 0 and > 0 there exists a constant Ck,(r) > 0 such that Z k jz1 − z2j dA(z1) k 2 ≤ Ck,(r)ρ(z2) ; for any z2 2 C: r exp (d (z ; z ) ) ρ(z ) C nD (z2) ' 1 2 1 Moreover, Ck,(r) ! 0 as r ! 1, for every k ≥ 0 and > 0.
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