Fredholmness of Some Toeplitz Operators

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Fredholmness of Some Toeplitz Operators Beyaz Başak Koca Istanbul University, Istanbul, Turkey IWOTA, 2017

Preliminaries Definition (Fredholm operator) Let H be a Hilbert space and let T B(H). T is said to be Fredholm operator if the range of T is closed, dimkert and dimkert are finite.

Preliminaries Definition (Fredholm operator) Let H be a Hilbert space and let T B(H). T is said to be Fredholm operator if the range of T is closed, dimkert and dimkert are finite. Theorem (Atkinson s characterization) Let H be a Hilbert space and let T B(H). Then T is Fredholm operator if and only if T + K(H) is invertible in the quotient algebra B(H)/K(H), where K(H) is the ideal of all compact operators on H.

A Banach algebra is a complex normed algebra A which is complete (as a topological space) and satisfies ST S T for all S, T A.

A Banach algebra is a complex normed algebra A which is complete (as a topological space) and satisfies ST S T for all S, T A. A C -algebra is a Banach algebra A with conjugate-linear involution which is an anti-isomorphism, that is, for all S, T A and λ in C and additional norm condition (λs + T ) = λs + T, (ST ) = T S, (S ) = S S S = S 2 for all S A.

If B is a subset of a C -algebra A, we set B = {T : T B} and if B = B we say B is self-adjoint. A closed self-adjoint subalgebra B of A is a C -subalgebra of A.

If B is a subset of a C -algebra A, we set B = {T : T B} and if B = B we say B is self-adjoint. A closed self-adjoint subalgebra B of A is a C -subalgebra of A. Theorem (Gelfand-Naimark Theorem) Any C -algebra is isometrically *-isomorphic to a C -subalgebra of B(H) for some Hilbert space H.

If B is a subset of a C -algebra A, we set B = {T : T B} and if B = B we say B is self-adjoint. A closed self-adjoint subalgebra B of A is a C -subalgebra of A. Theorem (Gelfand-Naimark Theorem) Any C -algebra is isometrically *-isomorphic to a C -subalgebra of B(H) for some Hilbert space H. For every subset C of A there is a smallest C -subalgebra of A containing C, called the C -subalgebra generated by C.

If B is a subset of a C -algebra A, we set B = {T : T B} and if B = B we say B is self-adjoint. A closed self-adjoint subalgebra B of A is a C -subalgebra of A. Theorem (Gelfand-Naimark Theorem) Any C -algebra is isometrically *-isomorphic to a C -subalgebra of B(H) for some Hilbert space H. For every subset C of A there is a smallest C -subalgebra of A containing C, called the C -subalgebra generated by C. If A is a C -algebra, then its commutator ideal I is the smallest (norm) closed, two-sided ideal of A containing {AB BA : A, B A}.

C -algebras generated by a system of unilateral weighted shifts Let n be a fixed positive integer. I = (i 1,..., i n ) be a multi-index of integers. I 0: i j 0 for all j = 1,..., n I = i 1 +... + i n For I 0 where z = (z 1,..., z n ) C n. z I = z i 1 1... z in n, Let ε k = (δ 1k,..., δ nk ) be an another multi-index, where δ ij is the Kronocker symbol. For the multi-index I I ε k = (i 1,..., i k 1,..., i n )

Let {e I } be an orthonormal basis of a complex Hilbert space H and let {w I,j : j = 1,..., n} be a bounded set of complex numbers such that for all I and 1 k, t n. Definition (Jewell, Lubin) w I,k w I +εk,t = w I,t w I +εt,k A system of unilateral weighted shifts is a family of n-operators A = {A 1,..., A n } on H such that A j e I = w I,j e I +εj, I 0, j = 1,..., n.

With aid of the positive {w I,j : j = 1,..., n} we define a set {β I } I 0 such that β I +εj = w I,j β I ; β 0 = 1.

With aid of the positive {w I,j : j = 1,..., n} we define a set {β I } I 0 such that β I +εj = w I,j β I ; β 0 = 1. Then, the space H 2 (β) = f (z) = f I z I : f I 2 βi 2 <. I 0 I 0

With aid of the positive {w I,j : j = 1,..., n} we define a set {β I } I 0 such that β I +εj = w I,j β I ; β 0 = 1. Then, the space H 2 (β) = f (z) = f I z I : f I 2 βi 2 <. I 0 I 0 is a Hilbert space with the inner product f, g = I 0 f I g I β 2 I and { zi β I } I 0 is an orthonormal basis for H 2 (β) (Jewell-Lubin).

Consider such {β I } I 0 that the multi-variable moment problem βi 2 = r 2i 1 1 r 2i 2 2... rn 2in dν(r 1, r 2,..., r n ) [0,1] n has a solution for these βi 2 s, that is, there exists a positive Borel measure ν defined on [0, 1] n for these βi 2 s satisfying above equality.

Consider such {β I } I 0 that the multi-variable moment problem βi 2 = r 2i 1 1 r 2i 2 2... rn 2in dν(r 1, r 2,..., r n ) [0,1] n has a solution for these βi 2 s, that is, there exists a positive Borel measure ν defined on [0, 1] n for these βi 2 s satisfying above equality. Let Ω denote the family of the systems A such that βi 2 s corresponding to the system A satisfying above property (i.e., the multi-variable moment problem has a solution for βi 2 s)

Consider such {β I } I 0 that the multi-variable moment problem βi 2 = r 2i 1 1 r 2i 2 2... rn 2in dν(r 1, r 2,..., r n ) [0,1] n has a solution for these βi 2 s, that is, there exists a positive Borel measure ν defined on [0, 1] n for these βi 2 s satisfying above equality. Let Ω denote the family of the systems A such that βi 2 s corresponding to the system A satisfying above property (i.e., the multi-variable moment problem has a solution for βi 2 s) and let ν A denote the measure corresponding to the system A.

L 2 ( D n, µ) (= L 2 (µ)) : the space of complex-valued functions on D n which are Lebesgue measurable and square-integrable with respect to the measure µ. Here µ is given on D n by dµ = 1 (2π) n dν(r 1, r 2,..., r n )dθ 1 dθ 2... dθ n (0 < θ i 2π)

L 2 ( D n, µ) (= L 2 (µ)) : the space of complex-valued functions on D n which are Lebesgue measurable and square-integrable with respect to the measure µ. Here µ is given on D n by dµ = 1 (2π) n dν(r 1, r 2,..., r n )dθ 1 dθ 2... dθ n (0 < θ i 2π) { 1 If A Ω, then is an orthonormal system in L 2 (µ). (2π) n } z I β I I 0

L 2 ( D n, µ) (= L 2 (µ)) : the space of complex-valued functions on D n which are Lebesgue measurable and square-integrable with respect to the measure µ. Here µ is given on D n by dµ = 1 (2π) n dν(r 1, r 2,..., r n )dθ 1 dθ 2... dθ n (0 < θ i 2π) { 1 If A Ω, then is an orthonormal system in L 2 (µ). (2π) n } z I β I I 0 Let H 2 ( D { n, µ) (= H 2 (µ)) be the subspace generated by the orthonormal 1 system L 2 (µ). (2π) n } z I β I I 0

Functional Model If A Ω, then the system A of unilateral weighted shifts on H and the system A z (= {A z1,..., A zn }) of the multiplication operators A zi on H 2 (µ) by the independent variables z j s, i = 1, 2,..., n are unitarily equivalent.

For simplicity, we consider n = 2. Let be S 1 = {(r 1, r 2 ) [0, 1] [0, 1] : r1 2 + r2 2 1} S 1 = {(r 1, r 2 ) [0, 1] [0, 1] : r1 2 + r2 2 = 1}. Let Ω 1 be the subset of Ω defined by Ω 1 = {A Ω : suppν A S 1, ν A (U(a)) > 0 for arbitrary neighborhood U(a) of each point a S 1 } Theorem If A Ω 1, then H 2 (µ) is a functional Hilbert space.

Theorem (Ergezen, Sadik) Let A Ω. A necessary and sufficient condition for the operator algebra generated by the system A to be isometrically isomorphic to the ball algebra is that A is in Ω 1.

Theorem (Ergezen, Sadik) Let A Ω. A necessary and sufficient condition for the operator algebra generated by the system A to be isometrically isomorphic to the ball algebra is that A is in Ω 1. Theorem (K., Sadik) Let A Ω. If the algebra generated by the system A is the ball algebra, then the commutator ideal of the C - algebra C (A) generated by the system A is the ideal of all compact operators K and C (A) = {T ψ + K : ψ C(suppµ), K K}. The quotient C (A)/K is -isomorphic to C(S 3 ), where S 3 is the unit sphere.

Definition If P denotes the orthogonal projection from L 2 (µ) onto H 2 (µ), then for ψ C(suppµ) the Toeplitz operator T ψ on H 2 (µ) with continuous symbol ψ is defined by T ψ f = P(ψf ) for f H 2 (µ).

Definition If P denotes the orthogonal projection from L 2 (µ) onto H 2 (µ), then for ψ C(suppµ) the Toeplitz operator T ψ on H 2 (µ) with continuous symbol ψ is defined by T ψ f = P(ψf ) for f H 2 (µ). Corollary Let ψ C(suppµ). Then The Toeplitz operator T ψ C (A) is Fredholm if and only if ψ(z) 0 for all z S 3.

Let Ω 2 be the subset of Ω defined by Ω 2 = {A Ω : ν A (U(1, 1)) > 0 for arbitrary neighborhood U(1, 1) of the point (1, 1) [0, 1] 2 }. Theorem If A Ω 2, then H 2 (µ) is a functional Hilbert space.

Theorem (Ergezen, Sadik) Let A Ω. A necessary and sufficient condition for the operator algebra generated by the system A to be isometrically isomorphic to the polydisc algebra is that A is in Ω 2.

Theorem (Ergezen, Sadik) Let A Ω. A necessary and sufficient condition for the operator algebra generated by the system A to be isometrically isomorphic to the polydisc algebra is that A is in Ω 2. Theorem (K., Sadik) Let A Ω 2. Then, the commutator ideal J of C (A) properly contains the ideal K of compact operators on H 2 (D 2 ). The quotient space J /K(H 2 (D 2 )) is isometrically isomorphic to C(T {0, 1}) K(H 2 (D)), where T is the unit circle and {0, 1} is the two-point space.

We assume that the measure ν A has of the form ν A (r 1, r 2 ) = ν 1 (r 1 )ν 2 (r 2 ), where both measures ν 1 and ν 2 are defined on [0, 1] and satisfying ν i (a, 1] > 0, i = 1, 2 for all 0 < a < 1. The measure µ A is then written as. µ A = 1 (2π) 2 ν 1ν 2 dθ 1 dθ 2

We assume that the measure ν A has of the form ν A (r 1, r 2 ) = ν 1 (r 1 )ν 2 (r 2 ), where both measures ν 1 and ν 2 are defined on [0, 1] and satisfying ν i (a, 1] > 0, i = 1, 2 for all 0 < a < 1. The measure µ A is then written as. Theorem (K., Sadik) µ A = 1 (2π) 2 ν 1ν 2 dθ 1 dθ 2 Let ψ C(suppµ). Then a necessary and sufficient condition for T ψ to be Fredholm is that ψ does not vanish in T 2 and ψ T 2 is homotopic to a constant.

An application for the unit ball case π: the quotient homomorphism from B(H) to B(H)/K(H) L: a subalgebra of B(H) such that the image π(l) is a commutative subalgebra of B(H)/K(H) S: an automorphism in the algebra π(l) such that Sπ(B) = π(b )S, where B, B L, that is, if B L, then SBS 1 = B + K, B L and K K(H).

An application for the unit ball case π: the quotient homomorphism from B(H) to B(H)/K(H) L: a subalgebra of B(H) such that the image π(l) is a commutative subalgebra of B(H)/K(H) S: an automorphism in the algebra π(l) such that Sπ(B) = π(b )S, where B, B L, that is, if B L, then SBS 1 = B + K, B L and K K(H). Consider the operator T = B 1 + B 2 S + K, where B 1, B 2 L ve K K.

An application for the unit ball case π: the quotient homomorphism from B(H) to B(H)/K(H) L: a subalgebra of B(H) such that the image π(l) is a commutative subalgebra of B(H)/K(H) S: an automorphism in the algebra π(l) such that Sπ(B) = π(b )S, where B, B L, that is, if B L, then SBS 1 = B + K, B L and K K(H). Consider the operator T = B 1 + B 2 S + K, where B 1, B 2 L ve K K. Theorem (K., Sadik) If the operator π(b 1 )π(b 1 ) π(b 2)π(B 2 ) has an inverse in π(l), then T = B 1 + B 2 S + K is Fredholm.

Take A Ω 1. Let L = C (A). Consider the operator T = T ψ1 + T ψ2 S + K, where T ψ1 ve T ψ2 are Toeplitz operators in C (A) with the symbols ψ 1, ψ 2 C(suppµ), respectively and Sf (w 1, w 2 ) = f (w 2, w 1 ) for all f H 2 (µ).

Take A Ω 1. Let L = C (A). Consider the operator T = T ψ1 + T ψ2 S + K, where T ψ1 ve T ψ2 are Toeplitz operators in C (A) with the symbols ψ 1, ψ 2 C(suppµ), respectively and Sf (w 1, w 2 ) = f (w 2, w 1 ) for all f H 2 (µ). The equation Tf = ϕ is written of the form K(z, w)ψ 1 (w 1, w 2 )f (w 1, w 2 )dµ(w 1, w 2 )+ B 4 K(z, w)ψ 2 (w 1, w 2 )f (w 2, w 1 )dµ(w 1, w 2 ) + (Kf )(z 1, z 2 ) = ϕ(z 1, z 2 ) B 4

Theorem (K., Sadik) If ψ 1 (z 1, z 2 )ψ 1 (z 2, z 1 ) ψ 2 (z 1, z 2 )ψ 2 (z 2, z 1 ) does not vanish in S 3, then all of Noether s theorems is true for the equation Tf = ϕ.

Theorem (K., Sadik) If ψ 1 (z 1, z 2 )ψ 1 (z 2, z 1 ) ψ 2 (z 1, z 2 )ψ 2 (z 2, z 1 ) does not vanish in S 3, then all of Noether s theorems is true for the equation Tf = ϕ.in particular, if we m+1 take w I,1 = 2+m+n, w n+1 I,2 = 2+m+n and Sf (w 1, w 2 ) = f (w 2, w 1 ) then the equation above has the form ψ 1 (w 1, w 2 )f (w 1, w 2 ) S 3 (1 z 1 w 1 z 2 w 2 ) 2 ds + ψ 2 (w 1, w 2 )f (w 2, w 1 ) S 3 (1 z 1 w 1 z 2 w 2 ) 2 ds+ +(Kf )(z 1, z 2 ) = ϕ(z 1, z 2 ), where ds is the surface measure in S 3.

References Coburn, L. A., 1969, The C -algebra generated by an isometry II, Trans. Amer. Math. Soc., 137,211-217. Coburn, L. A., 1973/74, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J., 23, 433-439. Douglas, R. G. and Howe, R., 1971, On the C -algebra of Toeplitz operators on the quarter plane, Trans. Amer. Math. Soc., 158, 203-217. Jewell, N. P. and Lubin, A. R., 1979, Commuting weighted shifts and analytic function theory in several variables, J. Operator Theory, 1, no.2, 207-223.

References Ergezen, F., 2008, On unilateral weighted shifts in noncommutative operator theory., Topology Appl., 155, no. 17-18, 1929-1934. Ergezen, F. and Sadik, N., 2010, On some operator algebras generated by unilateral weighted shifts., Publ. Math. Debrecen 76, no. 1-2, 21-30. Koca, B.B. and Sadik, N., 2012, C -Algebras Generated by a System of Unilateral Weighted Shifts and Their Application, Journal of Function Spaces and Applications, vol. 2012, 5 pages Koca, B.B., 2016, Fredholmness of Toeplitz operators on generalized Hardy spaces over the polydisc, Arch. Math., vol.107, pp.265-270.

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