Similarity Orbits And Unitary Orbits In The Calkin Algebra

Throughout this paper, C denotes the complex plane and H will always denote a complex separable infinite dimensional Hilbert space. Let B(H) denote the algebra of all bounded linear operators on H. We denote by K(H) the ideal of compact operators in B(H) and byÏ€the quotient map of B(H) onto the Calkin algebra A(H) (?) B(H)/K(H). If T∈B(H), then we also write (?) to denoteÏ€(T).The classification of operators is in a key position in operator theory. Similarity and unitary equivalence are two important equivalence relations on B(H). Given T∈B(H), the equivalence class containing T with respect to unitary equivalence (similarity), denoted by U(T)(S(T)), is called the unitary orbit (similarity orbit, respectively) of T. Voiculescu's non-commutative Weyl-von Neumann theorem and the similarity orbit theorem of Apostol, Fialkow, Herrero and Voiculescu help people to obtain deeper and better understanding of the structure of unitary orbits and similarity orbits in B(H).Since Calkin algebra is the unique nontrivial quotient algebra of B(H), it is natural to consider the classification in Calkin algebra. This will help to study the classification of operators. Corresponding to the case of operators, similarity and unitary equivalence in Calkin algebra can be defined similarly. For T∈B(H), we denote by S(?) and U(?) the similarity orbit and unitary orbit of (?) respectively. Given A, B∈B(H), (?),(?) are said to be strongly similar (Strongly unitarily equivalent), if A is similar(unitarily equivalent) to some compact perturbation of B; in this case, A, B are said to be similar modulo compact(unitarily equivalent modulo compact), denoted by A (?) B(A (?) B). It is obvious that this defines two equivalence relations on Calkin algebra, strongly similarity implies similarity and strongly unitary equivalence implies unitary equivalence. The strong similarity orbit and strongly unitary orbit of (?) areÏ€[S(T)] andÏ€[U(T)] respectively.Normal elements in the Calkin algebra have been completely classified by Brown, Douglas and Fillmore by virtue of the theory of C^*-algebra extensions. However, unitary orbits of other elements in A(H) are far less understood. For example, it follows immediately from B-D-F Theorem that, for normal elements in the Calkin algebra, strongly unitary equivalence equals unitary equivalence. However, this is not necessarily true for other elements in the Calkin algebra.On the other hand, it is an immediate consequence of B-D-F Theorem that the unitary orbits of normal elements are necessarily closed. What about other unitary orbits in the Calkin algebra? Apostol, Fialkow, Herrero and Voiculescu attempted to investigate related questions by studying the number of the strongly unitary orbits in a unitary orbit. Given T∈B(H), let CU_e(T) denote the number of the strongly unitary orbits in U(?). They raised the following conjecture in their monograph "Approximation of Hilbert space operators, vol.â…¡".Conjecture AFHV-â…  If T∈B(H), then CU_e(T) <∞.Davidson proved that if T is Kakutanrs weighted shift operator, then CU_e(T) =∞and U(?) is not closed, thus gave a negative answer to Conjecture AFHV-â… . However, does CU_e(T) <∞imply that U(?) is closed? In particular, does U(?) =Ï€[U(T)] imply that U(?) is closed? This is one of the questions that we shall investigate.For nonnormal elements in B(H) and A(H), more often than not, people consider similarity. In view of B-D-F Theorem, it is natural to investigate whether or not strong similarity equals similarity in the Calkin algebra. More precisely, given an operator T on H, does S[(?)] coincide withÏ€[S(T)]?Fialkow proved that if N∈B(H) is normal, then S(?) =Ï€[S(N)]. Apostol and Voiculescu first found that there exist some Jordan operators T satisfying that S(?)≠π[S(T)] and S(?) is not connected.If k∈N and {e₁, e₂,…,e_k} is the canonical ONB of C^k. then define the operator qk on C^k as An operator T on H is called a Jordan operator, if T is unitarily equivalent to (?)where m∈N,λ₁,λ₂,…,λ_m are pairwise distinct, n_j∈N, 0≤α_kj≤∞.(?); if, in addition, there exists at most one k, 1≤k≤n_j, such thatα_kj＝∞for each j, then T is called a nice Jordan operator.Moreover, Apostol, Fialkow, Herrero and Voiculescu proved that if T is not similarto any compact perturbation of any Jordan operator, then (?). In view of this result, they raised the following conjecture.Conjecture AFHV-â…¡ If T∈B(H) is not similar to a compact perturbation of a Jordan operator, then S(?) =Ï€[S(T)].In 2005, Y. Q. Ji, P. H. Wang and X. J. Xu made a breakthrough in the study of this conjecture. They introduced a new class of operators on H, that is ES(H). T∈B(H) is called an ES operator, if T is similar to a compact perturbation of(?). Y. Q. Ji, P. H. Wang and X. J. Xu proved that (?) and for A,B∈ES(H), (?), (?) are similar if and only if (?), (?) are strongly similar. This result seems to suggest that the answer to Conjecture AFHV-II is positive.It is not difficult to see that the study of Conjecture AFHV-â… and AFHV-â…¡are reduced to studying CS_e(T) and CU_e(T) respectively for T∈B(H). where CS_e(T) denotes the number of the strong similarity- orbits in S(?). For n∈N∪{∞}, denote C_e(n) = {T∈B(H) : CS_e(T) = n}. Let n∈N. T∈B(H) is called an ES_n operator (EU_n operator), denoted by T∈ES_n(H)(T∈EU_n(W)), if T is similar (unitarily equivalent) to some compact perturbation of (?). If (?) and (?). then we denote gcd (?)= sup{n∈N : n divides any element in (?)}. If T∈B(H), then denote GCD(T) = gcd{ind (λ- T) : (?)}. If (?), then we denote by intεthe interior ofε. Given two operators A, B on H, ifσ_lre(A) =σ_lre(B), ind (λ- A) = ind (λ- B). (?), then we shall denote (?).The first part of this thesis mainly deals with similarity orbits and unitary orbits in the Calkin algebra. Given T∈B(H), we give a characterization of S(?). Furthermore we obtain the following result which characterizes CS_e(T).Theorem 0.1 If T∈B(H), then CS_e(T) = min{i∈N : T∈ES_i(H)} = gcd{ind X : X is a Fredholm operator and (?)}, where min(?) . Corollary 0.2 C_e(1) = ES(H). More precisely, if T∈B(H), then the following three are equivalent:Based on the results above, we obtain the following result which gives a negative answer to Conjecture AFHV-â…¡.Theorem 0.3 (i) Given n∈N and a nonempty compact subsetσof C, there exists an A in B(H) such that CS_e(A) = n andσ(A) =σ.(ii) If T∈B(H), n∈N andσ_e(T) =σ_lre(T), then n divides GCD(T) if and only if there exists R∈B(H) such that CS_e(R) = n and (?). If, in addition,σ_e(T) is perfect, then n divides GCD(T) if and only if there exists R∈B(H) suchthat CS_e(R) = n and (?).(iii) If T∈B(H), n∈N andσ_lre(T)=αΩ, whereΩin a simply connected analytic Cauchy domain in C, then n divides GCD(T) if and only if there exists R∈B(H)such that CS_e(R) = n and (?).We describe the size of the classes ES_n(H)(n∈N) and C_e(n)(n∈N∪{∞}) by characterizing their closures and interiors.Theorem 0.4 Let n∈N. Then(i) If T∈B(H) and each component ofσ_e(T)\σ_lre(T) is simply connected, then T∈(?) if and only if n divides GCD(T);(ii) (?) = B(H) and {X∈B(H) : GCD(X) divides n} (?) int ES_n(H);(iii) (?)= B(H) and hence (?) C_e(i) is nowhere dense in B(H). Corollary 0.5 The interior of (?) is dense in A(H) and hence (?) is nowhere dense in A(H).We characterize those connected similarity orbits in the Calkin algebra.Theorem 0.6 Let T∈B[H). Then(i) S(?) is connected if and only if CS_e(T) = 1 or T is not similar to any compact perturbation of any nice Jordan operator;(ii) (?) is arcwise connected.Given T∈B(H), we give a characterization of U(?); furthermore we characterize CU_e(T).Theorem 0.7 If T∈B(H), then CU_e(T) = min{i∈N : T∈EU_i(H)} = gcd{ind X : (?) is unitary in A(H) and (?)}, where min(?).Using the preceding results, we prove that, given a cardinal 1≤n≤N₀, there exists an operator T on H such that CU_e(T) = n and (?). We study Conjecture AFHV-â…  in a different direction from Davidson and obtain the following result which also gives a negative answer to Conjecture AFHV-â… .Theorem 0.8 Given T∈B(H), n∈N andε> 0, there exist an operator T₁∈B(H) such that ||T₁ - T|| <ε, CU_e(T₁) =∞and T₁∈int {X∈B(H) : n + 1 divides CU_e(X)}.Corollary 0.9 (i) The set (?) is dense in A(H).(ii) (?) is nowhere dense in A(H) for all n∈N. In particular. (?) is nowhere dense in A(H) for all n∈N.The other part of our research focuses on the classification of operators with respect to strongly approximative similarity. If T∈B(H),σis a clopen subset ofσ(T), we denote by E(σ; T) the Riesz idempotent operator of T with respect toσand by H(σ; T) the corresponding Riesz spectral space, i.e. H(σ;T) = ran E(σ;T). In particular, we write H(λ;T) instead of H({λ};T). Two operators A,B on H are said to be strongly approximative similar, if, givenε> 0. there exist compact operators K₁,K₂ such that (i) ||K_i|| <ε(i = 1,2); (ii) A + K₁ B + K₂; (iii)σ₀(A) =σ₀(B) and dim H(λ; A) = dim H(λ; B) for allλ∈σ₀(A).We introduce two classes of operators on H,i.e., QT(H) and Nic(H), which are dense in quasitriangular operators and B(H) respectively. We obtain the following theorem which implies that strongly approximative similarity is an equivalence relation on QT(H)∪Nic(H) and give a classification of operators in QT(H)∪Nic(H) with respect to strongly approximative similarity.Theorem 0.10 If A, B∈QT(H)∪Nic(H), then (?) B if and only ifσ(A) =σ(B) and (?) for any nonempty clopen subsetσofσ(A).As a corollary we obtain a classification of a class of Cowen-Douglas operators.Corollary 0.11 Let A,B∈B(H) be two Cowen-Douglas operators. If each component (?)(A) satisfies that int (?), then (?) B if and only if (?).