A rank formula for the self-commutators of rational Toeplitz tuples

Transcription

1 wang et al Journal of Inequalities and Applications 2016) 2016:184 DOI /s x R E S E A R C Open Access A rank formula for the self-commutators of rational Toeplitz tuples In Sung wang 1,AnyunKim 2* and Sumin Kim 1 * Correspondence: ahkim@changwonackr 2 Department of Mathematics, Changwon National University, Changwon, , Korea Full list of author information is available at the end of the article Abstract In this paper we derive a rank formula for the self-commutators of tuples of Toeplitz operators with matrix-valued rational symbols MSC: Primary 47B20; 47B35; 47A13; secondary 3010; 47A57 Keywords: block Toeplitz operators; jointly hyponormal; bounded type functions; rational functions; self-commutators 1 Introduction Let and K be complex ilbert spaces, let B, K) be the set of bounded linear operators from to K, andwriteb):=b, ) For A, B B), we let A, B:=AB BA An operator T B) is said to be normal if T, T=0,hyponormalifT, T 0 For an operator T B), we write ker T and ran T for the kernel and the range of T,respectively For a subset M of a ilbert space, cl M and M denote the closure and the orthogonal complement of M,respectivelyAlso,letT D be the unit circle where D denotes the open unit disk in the complex plane C) Recall that L L T) isthesetofbounded measurable functions on T, that the ilbert space L 2 L 2 T) has a canonical orthonormal basis given by the trigonometric functions e n z)=z n, for all n Z, and that the ardy space 2 2 T) is the closed linear span of {e n : n 0}Anelementf L 2 is said to be analytic if f 2 Let := L 2, ie, is the set of bounded analytic functions on D We review the notion of functions of bounded type and a few essential facts about ankel and Toeplitz operators and for that we will use 1 4 For ϕ L,wewrite ϕ + Pϕ 2 and ϕ P ϕ z 2, where P and P denote the orthogonal projection from L 2 onto 2 and 2 ),respectively Thus we may write ϕ = ϕ + ϕ + We recall that a function ϕ L is said to be of bounded type or in the Nevanlinna class N )iftherearefunctionsψ 1, ψ 2 such that ϕz)= ψ 1z) ψ 2 z) for almost all z T 2016 wang et al This article is distributed under the terms of the Creative Commons Attribution 40 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made

2 wangetal Journal of Inequalities and Applications 2016) 2016:184 Page 2 of 14 We recall 5, Lemma 3, that if ϕ L then ϕ is of bounded type ker ϕ {0} 11) Assume now that both ϕ and ϕ are of bounded type Then from the Beurling s theorem, ker ϕ = θ 0 2 and ker ϕ+ = θ + 2 for some inner functions θ 0, θ + Wethushave b := ϕ θ 0 2, and hence we can write ϕ = θ 0 b and similarly ϕ + = θ + a for some a 2 12) By Kronecker s lemma 3, p183, if f then f is a rational function if and only if rank f <, which implies that f is rational f = θb with a finite Blaschke product θ 13) Let M n r denote the set of all n r complex matrices and write M n := M n n ForX a ilbert space, let L 2 X L2 X T) betheilbertspaceofx -valued norm square-integrable measurable functions on T and let L X L X T) bethesetofx-valued bounded measurable functions on T WealsoletX 2 2 X T) be the corresponding ardy space and X X T)=L X 2 X WeobservethatL2 C n = L2 C n and C 2 n = 2 C n For a matrix-valued function ϕ ij ) L M n,wesaythat is of bounded type if each entry ϕ ij is of bounded type, and we say that is rational if each entry ϕ ij is a rational function Let ϕ ij ) L M n be such that is of bounded type Then each ϕ ij is of bounded type Thus in view of 12), we may write ϕ ij = θ ij b ij,whereθ ij is inner and θ ij and b ij are coprime, in other words, there does not exist a nonconstant inner divisor of θ ij and b ij Thusifθ is theleastcommonmultipleof{θ ij : i, j =1,2,,n},thenwemaywrite =ϕ ij )=θ ij b ij )=θa ij ) θa where A aji ) 2 M n ) 14) In particular, Aα) is nonzero whenever θα)=0and α <1 For ϕ ij L M n,wewrite + := Pϕ ij ) 2 M n and := P ϕ ij ) 2 Mn Thus we may write = + + owever, it will often be convenient to allow the constant term in ence,ifthereisnoconfusionwemayassumethat shares the constant term with + :inthiscase, 0) = + 0) + 0) If = + + L M n is such that and are of bounded type, then in view of 14), we may write + = θ 1 A and = θ 2 B, 15) where θ 1 and θ 2 are inner functions and A, B M 2 n Inparticular,if L M n is rational then the θ i canbechosenasfiniteblaschkeproducts,asweobservedin13) For simplicity, we write 0 2 for z2 M n

3 wangetal Journal of Inequalities and Applications 2016) 2016:184 Page 3 of 14 We now introduce the notion of ankel operators and Toeplitz operators with matrixvalued symbols If is a matrix-valued function in L M n,thent : C 2 n 2 C n denotes Toeplitz operator with symbol defined by T f := P n f ) forf 2 C n, where P n is the orthogonal projection of L 2 C n onto 2 Cn A ankel operator with symbol L M n is an operator : C 2 n 2 Cn defined by f := J n P n f ) forf 2 C n, where Pn is the orthogonal projection of L2 C n onto 2 C n) and J n denotes the unitary operator from L 2 C n onto L2 C n given by J nf )z):=zf z)forf L 2 C nfor L M n m,write z):= z) A matrix-valued function M n m is called inner if = I m almost everywhere on T, where I m denotes the m m identity matrix If there is no confusion we write simply I for I m The following basic relations can easily be derived: T = T, = L Mn ) ; 16) T T T =, L Mn ) ; 17) T =, = T L Mn, M n ) 18) In 2006, Gu et al 6 have considered the hyponormality of Toeplitz operators with matrix-valued symbols and characterized it in terms of their symbols Lemma 11 yponormality of block Toeplitz operators 6) For each L M n, let E ):= { K M n : K 1 and K M n } Then T is hyponormal if and only if is normal and E ) is nonempty For a matrix-valued function M 2 n r,wesaythat M 2 n m is a left inner divisor of if is an inner matrix function such that = A for some A M 2 m r Wealsosaythat twomatrixfunctions M 2 n r and M 2 n m are left coprime if the only common left inner divisor of both and is a unitary constant, and that M 2 n r and M 2 m r are right coprime if and are left coprime Two matrix functions and in M 2 n are said to be coprime if they are both left and right coprime We note that if M 2 n is such that det 0, then any left inner divisor of is square, ie, M 2 n cf 7) If M 2 n is such that det 0,thenwesaythat M 2 n is a right inner divisor of if is a left inner divisor of Let { i M n : i J} be a family of inner matrix functions The greatest common left inner divisor d and the least common left inner multiple m of the family { i M n :

4 wangetal Journal of Inequalities and Applications 2016) 2016:184 Page 4 of 14 i J} are the inner functions defined by d C 2 p = i C 2 n and mc 2 q = i C 2 n i J i J Similarly, the greatest common right inner divisor d and the least common right inner multiple m of the family { i M n : i J} are the inner functions defined by d 2 C r = i C 2 n and m 2 C s = i C 2 n i J i J The Beurling-Lax-almos theorem guarantees that d and m exist and are unique up to a unitary constant right factor, and d and m are unique up to a unitary constant left factor We write d =left-gcd{ i : i J}, d = right-gcd{ i : i J}, m =left-lcm{ i : i J}, m = right-lcm{ i : i J} If n = 1,then left-gcd{ } = right-gcd{ } simply denoted gcd{ }) and left-lcm{ } = right-lcm{ } simply denoted lcm{ }) In general, it is not true that left-gcd{ } = right-gcd{ } and left-lcm{ } = right-lcm{ } If θ is an inner function we write I θ for θi n and Zθ) for the set of all zeros of θ Lemma 12 Let i := I θi for an inner function θ i i J) a) left-gcd{ i : i J} = right-gcd{ i : i J} = I θd, where θ d = gcd{θ i : i J} b) left-lcm{ i : i J} = right-lcm{ i : i J} = I θm, where θ m = lcm{θ i : i J} Proof See 7, Lemma 21 In view of Lemma 12,if i = I θi for an inner function θ i i J), we can define the greatest common inner divisor d and the least common inner multiple m of the i by and d gcd{ i : i J} := I θd, whereθ d = gcd{θ i : i J} m lcm{ i : i J} := I θm, whereθ m = lcm{θ i : i J} Both d and m are diagonal-constant inner functions, ie, diagonal inner functions, and constant along the diagonal By contrast with scalar-valued functions, in 14), I θ and A need not be right) coprime If =left-gcd{i θ, A} in the representation 14), that is, = θa, then I θ = l and A = A l for some inner matrix l where l 2 M n because deti θ ) 0) and some A l 2 M n Therefore if L M n is of bounded type then we can write = A l l, wherea l and l are left coprime 19)

5 wangetal Journal of Inequalities and Applications 2016) 2016:184 Page 5 of 14 In this case, A l l is called the left coprime factorization of and write, briefly, = A l l left coprime) 110) Similarly, we can write = r A r, wherea r and r are right coprime 111) In this case, r A r is called the right coprime factorization of and we write, succinctly, = r A r right coprime) 112) In this case, we define the degree of by deg ):=dim r ), where ):=C 2 n 2 Cn for an inner function Itwasknowncf 8, Lemma 33) that if θ is a finite Blaschke product then I θ and A M 2 n are left coprime if and only if they are right coprime In this viewpoint, in 110)and112), l or r is I θ θ afiniteblaschke product) then we shall write = θa coprime) On the other hand, we recall that an operator T B) is said to be subnormal if T has a normal extension, ie, T = N,whereN is a normal operator on some ilbert space K such that is invariant for N The Bram-almos criterion for subnormality 9, 10 states that an operator T B) is subnormal if and only if i,j T i x j, T j x i ) 0 for all finite collections x 0, x 1,,x k It is easy to see that this is equivalent to the following positivity test: T, T T 2, T T k, T T, T 2 T 2, T 2 T k, T 2 0 allk 1) 113) T, T k T 2, T k T k, T k Condition 113) provides a measure of the gap between hyponormality and subnormality In fact the positivity condition 113) fork = 1 is equivalent to the hyponormality of T, while subnormality requires the validity of 113) for all k Fork 1, an operator T is said to be k-hyponormal if T satisfies the positivity condition 113) forafixedk Thus the Bram-almos criterion can be stated thus: T is subnormal if and only if T is k-hyponormal forall k 1 The notion of k-hyponormality has been considered by many authors aiming at understanding the bridge between hyponormality and subnormality In view of 113), between hyponormality and subnormality there exists a whole slew of increasingly stricter conditions, each expressible in terms of the joint hyponormality of the tuples I, T, T 2,,T k ) Given an n-tuple T =T 1,,T n ) of operators on, welet

6 wangetal Journal of Inequalities and Applications 2016) 2016:184 Page 6 of 14 T, T B )denotetheself-commutator of T,definedby T1 T, T, T 1 T2, T 1 Tn, T 1 T1 :=, T 2 T2, T 2 Tn, T 2 T1, T n T2, T n Tn, T n By analogy with the case n = 1,we shall say11, 12that T is jointly hyponormal or simply, hyponormal)ift, T 0, ie,t, T is a positive-semidefinite operator on Tuples T T 1,,T m ) of block Toeplitz operators T i i =1,,m) will be called a block) Toeplitz tuples Moreover, if each Toeplitz operator T i has a symbol i which is a matrix-valued rational function, then the tuple T T 1,,T m ) is called a rational Toeplitz tuple In this paper we will derive a rank formula for the self-commutator of a rational Topelitz tuple 2 The results and discussion For an operator S B), S B) is called the Moore-Penrose inverse of S if SS S = S, S SS = S, S S ) = S S, and SS ) = SS It is well known 13, Theorem 872, that if an operator S on a ilbert space has a closed range then S has a Moore-Penrose inverse Moreover, the Moore-Penrose inverse is unique whenever it exists On the other hand, it is well known that if S := A B B C on 1 2 where the j are ilbert spaces, A B 1 ), C B 2 ), and B B 2, 1 )), then S 0 A 0, C 0, and B = A 1 2 DC 1 2 for some contraction D; 21) moreover, in 14, Lemma 12, and 15, Lemma 21, it was shown that if A 0, C 0, and ran A is closed then S 0 B A B C and ran B ran A, 22) or equivalently 12, Lemma 14, Bg, f 2 Af, f Cg, g for all f 1, g 2 23) and furthermore, if both A and C are of finite rank then rank S = rank A + rank C B A B ) 24) In fact, if A 0andran A is closed then we can write A 0 0 ran A ran A A = :, 0 0 ker A ker A

7 wangetal Journal of Inequalities and Applications 2016) 2016:184 Page 7 of 14 so that the Moore-Penrose inverse of A is given by A A 0 ) 1 0 = 25) 0 0 Proposition 21 If A B) has a closed range then AA A) A is the orthogonal projection onto ran A Proof Suppose A B)hasaclosedrangeThen25)canbewrittenas P ran A AP ran A ) 1 = P ran A A P ran A 26) Since by assumption, A A has also a closed range, there exists the Moore-Penrose inverse A A) Observe A A A ) A ) A A A ) A ) = A A A ) A and A A A ) A ) = A A A ) A, which implies that AA A) A is an orthogonal projection Put K := ran A A = ran A =ker A) We then have A A A ) A = AP K A A ) PK A = A P K A A ) P K ) 1A by 25) ), which implies that ranaa A) A )=ran A In the sequel we often encounter the following matrix: A A A B S :=, B A B, B where A has a closed range If S 0andifA and B, B are of finite rank then by 24), we have rank S = rank A A ) + rank B, B B A A A ) A B ) 27) Thus, if we write P K for the orthogonal projection onto K := ran A,thenbyProposition21 we have rank S = rank A ) + rank B, B B P K B ) = rank A ) + rank B P K B BB ) 28)

8 wangetal Journal of Inequalities and Applications 2016) 2016:184 Page 8 of 14 If, L M n,thenby17), T, T = + T Since the normality of is a necessary condition for the hyponormality of T cf 15), the positivity of is an essential condition for the hyponormality of T If L M n,thepseudo-self-commutator of T is defined by T, T p := Then T is said to be pseudo-hyponormal if T, T p 0 We also see that if L M n then T, T =T, T p + T Proposition 22 Let + + L M n be such that and are of bounded type Thus in view of 14), we may write + = θ 1 A and = θ 2 B, where θ 1 and θ 2 are inner functions and A, B 2 M n If T is hyponormal then θ 2 is an inner divisor of θ 1, ie, θ 1 = θ 0 θ 2 for some inner function θ 0 Proof See 7, Proposition 32 In view of Proposition 22, when we study the hyponormality of block Toeplitz operators with bounded type symbols ie, and are of bounded type) we may assume that the symbol + + L M n is of the form + = θ 0 θ 1 A and = θ 0 B, where θ 0 and θ 1 are inner functions and A, B M 2 n We first observe that if T =T ϕ, T ψ ) then the self-commutator of T can be expressed as T, T Tϕ =, T ϕ Tψ, T ϕ ϕ+ ϕ+ ϕ Tϕ, T ψ Tψ, T = ϕ ϕ + ψ+ ψ ϕ ψ ψ ϕ+ + ϕ ψ ψ + ψ+ 29) ψ ψ For a block Toeplitz pair T T, T ), the pseudo-commutator of T is defined by T, T T p :=, T p T, T p T, T p T, T p = Let i L M n i =1,2,,m) be normal and mutually commuting and let σ be a permutation on {1,2,,m} Then evidently, T := T 1,,T m ) is hyponormal T σ := T σ 1),,T σ m) ) is hyponormal 210)

9 wangetal Journal of Inequalities and Applications 2016) 2016:184 Page 9 of 14 Moreover, we have rank T, T = rank T σ, T σ 211) For every m 0 m,lett m0 := T 1,,T m0 ) Since T, T T =, T m0 m0, we can see that if T is hyponormal then in view of 210), every sub-tuple of T is hyponormal We then have the following Lemma 23 Let i L M n be normal and mutually commuting Let T T 1,,T m ) and S T 1 1,,T m m ), where the i are mutually commuting and are invertible constant normal matrices commuting with j and j for each i, j =1,2,,m Then T is hyponormal S is hyponormal Furthermore, rankt, T=rankS, S Proof In view of equation 210), it suffices to prove the lemma when i = I for all i = 2,,mPutT := T, TandS := S, S Since 1 is a constant normal matrix commuting with j, it follows that, for all j >1, S 1j = 1 1 ) + j ) + j ) 1 1 ) = 1 ) + 1 j ) + j ) 1 1 ) = T 1 1 ) + j ) + j ) T 1 1 ) = T 1 1 ) + j ) + j ) 1 1 ) = T 1 1 ) + j ) + j ) ) 1 ) = T 1 T 1j Observe that S 11 = 1 1 ) ) ) 1 1 ) = 1 ) ) ) 1 1 ) 1 = T 1 1 ) + 1 ) + T 1 T 1 1 ) 1 ) T 1 = T 1 1 ) + 1 ) + 1 ) 1 ) ) T 1 = T 1 T 11 T 1 Let Q be the block diagonal operator with the diagonal entries T 1, I,,I) Then Q is invertible and S = QT Q, which gives the result

10 wangetaljournal of Inequalities and Applications 2016) 2016:184 Page 10 of 14 Lemma 24 Let T T 1, T 2,T m ), where the i L M n i =1,,m) are normal and mutually commuting If S := T 1 j0, T 2,T m ) for some j 0 2 j 0 m), then T is hyponormal S is hyponormal Furthermore, rankt, T=rankS, S Proof Obvious Corollary 25 Let i L M n i =1,,m) be normal and mutually commuting Let T T 1,T m ) and put S := T 1 1 m, T 2 2 m,,t m 1 m 1 m, T m ), where the i i =1,,m 1)are mutually commuting and are invertible constant normal matrices commuting with j for each j =1,,m Then T is hyponormal S is hyponormal Furthermore, rankt, T=rankS, S Proof This follows from Lemmas 23 and 24 We now have the following Theorem 26 Let i M n i =1,2,,m 1)be mutually commuting and normal rational functions of the form i = A i i left coprime), where the i are inner matrix functions and m m ) + m) + L M n If T := T 1,,T m ) is hyponormal then rank T, T = deg )+rank T 1,, T m 1, m p, 212) where := right-lcm{ i : i =1,2,,m 1} and 1, m := m) + P 2 0 m) + ) Proof Let := 1,, m 1 ) Since i i ) + M n i = 1,2,,m 1),T is hyponormal if and only if T, T = m m T 0, m, T m or equivalently, for each X m 1 j=1 2 C n and Y 2 C n, m Y, X 2 X, X T m, T m Y, Y 213)

11 wangetaljournal of Inequalities and Applications 2016) 2016:184 Page 11 of 14 Since cl ran i = i )i =1,2,,n 1), it follows that m 1 m 1 cl ran = cl ran i = i )= i=1 i=1 m 1 ) i C 2 n i=1 = 2 C n ) = )=cl ran, 214) where ):=C 2 n 2 C nifthe i are rational functions then, by 13) and14), we can write i = θ i A i θ i, finite Blaschke product) Since i is a right inner divisor of I θi,wehavedeg i ) degi θi )=ndegθ i )< Thus since by 214), cl ran = )and deg )=rank = rank = deg )< Therefore is of finite rank and hence, so is and, moreover, rank ) = rank ) = rank )=deg ) Thus by 27), we have rank T, T = rank m m T m, T m = rank ) + rank T m, T m ) m ) m = deg )+rank T m, T m m ) m ) On the other hand, by Proposition 21, ) is the projection P )Therefore it follows from 17)and18)that T m, T m m ) m = T m, T m m m = m+ I ) m+ m m = m+ T ) T m+ ) m m = m+ m+ m m = T 1,, T m 1, m p, which gives the result Very recently, the hyponormality of rational Toeplitz pairs was characterized in 16

12 wangetaljournal of Inequalities and Applications 2016) 2016:184 Page 12 of 14 Lemma 27 yponormality of rational Toeplitz pairs) 16 Let T T, T ) be a Toeplitz pair with rational symbols, L M n of the form + = θ 0 θ 1 A, = θ 0 B, + = θ 2 θ 3 C, = θ 2 D coprime) 215) Assume that θ 0 and θ 2 are not coprime Assume also that Bγ 0 ) and Dγ 0 ) are diagonalconstant for some γ 0 Zθ 0 ) Then the pair T is hyponormal if and only if i) and are normal and = ; ii) = with := Bγ 0 )Dγ 0 ) 1 ); iii) T 1, is pseudo-hyponormal with := θ 0 θ 1 θ 3 θ, where θ := gcdθ 1, θ 3 ) and := left-gcdi θ0 θ, θθ 3 A θ 1 C )) We now get a rank formula for the self-commutators of Toeplitz m-tuples Corollary 28 For each i =1,2,,m, suppose that i = i ) + i) + L M n is a matrixvalued normal rational function of the form i ) + = θ i δ i A i and i ) = θ i B i coprime), where the θ i and the δ i are finite Blaschke products and there exists j 0 1 j 0 m) such that θ j0 and θ i are not coprime for each i = 1,2,,m Suppose i j = j i for all i, j = 1,,m Assume that each B i γ 0 ) is diagonal-constant for some γ 0 Zθ i ) If T T 1, T 2,,T m ) is hyponormal then rank T, T = deg )+rank T 1,, T 1, j j0 p, 0 where := right-lcm{θ i δ i δ j0 δi) i) : i =1,2,,m} ere δi):=gcd{δ i, δ j0 } and i):= left-gcd{θ i δi), δi)δ j0 A i δ i A j0 i) )} with i):=b i γ 0 )B j0 γ 0 ) 1 Proof Suppose T is hyponormal Since every sub-tuple of T is hyponormal, we can see that T i, T j ) is hyponormal for all i, j =1,2,,mInviewof210), we may assume that j 0 = mput S := T 1 1) m, T 2 2) m,,t m 1 m 1) m, T m ) It follows from Corollary 25 that T is hyponormal S is hyponormal Since δi)=gcd{δ i, δ m },wecanwrite δ i = δi)ω i and δ m = δi)ω m, where ω i is a finite Blaschke product for i = 1,2,,m Since i) = left-gcd{θ i δi), δi)δ m A i δ 1 A m i) )}, we get the following left coprime factorization: i i) m = ω m A i ω i i)a m) i) θi δ i δ m δi) i) Thus the result follows at once from Theorem 26

13 wangetaljournal of Inequalities and Applications 2016) 2016:184 Page 13 of 14 We conclude with the following Corollary 29 For each i = 1,2,,m, suppose that φ i = φ i ) +φ i ) + L is a rational function of the form φ i ) + = θ i a i and φ i ) = θ i b i coprime) If there exists j 0 1 j 0 m) such that θ j0 and θ i are not coprime for each i =1,2,,mand T T φ1, T φ2,,t φm ) is hyponormal then rank T, T = rank T j0, T j0 Proof For each i =1,2,,m, letλi) :=b i γ 0 )b j0 γ 0 ) 1 for some γ 0 Zθ i ) Write θi) gcd{θ i,a i a j0 λi))}sincet T φ1, T φ2,,t φn ) is hyponormal, T φi, T φj0 ) is hyponormal for all i =1,2,,n Thus it follows from Lemma 27 that T 1,ωi) φ is hyponormal with j 0 ωi):=θ i θi) Observe that φ 1,ωi) ) j 0 + = θi)c i and φ 1,ωi) j 0 ) = θ ib i coprime), where c i := P θi)) a i ) Since T 1,ωi) φ is hyponormal, it follows from Proposition 22 that θ i j 0 is an inner divisor of θi) and hence θi)=θ i Thus the result follows from Corollary 28 3 Conclusions The self-commutators of bounded linear operators play an important role in the study of hyponormal and subnormal operators The main result of this paper is to derive a rank formula for the self-commutators of tuples of Toeplitz operators with matrix-valued rational symbols This result will contribute to the study of Toeplitz operators and the bridge theory of operators Competing interests The authors declare that they have no competing interests Authors contributions The authors contributed equally and significantly in writing this paper Author details 1 Department of Mathematics, Sungkyunkwan University, Suwon, , Korea 2 Department of Mathematics, Changwon National University, Changwon, , Korea Acknowledgements The work of the first author was supported by National Research Foundation of Korea NRF) grant funded by the Ministry of Education, Science and Technology No ) The work of the second author was supported by Basic Science Research Program through the National Research Foundation of Korea NRF) funded by the Ministry of Education No 2015R1D1A3A ) Received: 14 April 2016 Accepted: 2 July 2016 References 1 Böttcher, A, Silbermann, B: Analysis of Toeplitz Operators Springer, Berlin 2006) 2 Douglas, RG: Banach Algebra Techniques in Operator Theory Academic Press, New York 1972) 3 Nikolskii, NK: Treatise on the Shift Operator Springer, New York 1986) 4 Peller, VV: ankel Operators and Their Applications Springer, New York 2003) 5 Abrahamse, MB: Subnormal Toeplitz operators and functions of bounded type Duke Math J 43, )

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