On a generalization of the Jordan canonical form theorem on separable Hilbert spaces

We prove a generalization of the Jordan canonical form theorem for a class of bounded linear operators on complex separable Hilbert spaces.


Introduction
Throughout this paper, all Hilbert spaces discussed are complex and separable while all operators are bounded and linear on the Hilbert spaces. Let H be a Hilbert space and let L (H ) be the set of bounded linear operators on H . An idempotent P is an operator such that P 2 = P . A projection Q is an idempotent such that Q = Q * . Unless otherwise stated, the term algebra will always refer to a unital subalgebra of L (H ) which is closed in the strong operator topology. An operator A on H is said to be irreducible if its commutant {A} ′ ≡ {B ∈ L (H ) : AB = BA} contains no projections other than 0 and I, introduced by P. Halmos [7]. (The separability assumption is necessary because on a non-separable Hilbert space every operator is reducible.) An operator A on H is said to be strongly irreducible if XAX −1 is irreducible for every invertible operator X in L (H ), introduced by F. Gilfeather [6]. Equivalently, an operator A is strongly irreducible if and only if {A} ′ contains no idempotents other than 0 and I.
On a finite dimensional Hilbert space K , the Jordan canonical form theorem shows that every operator can be uniquely written as a (Banach) direct sum of Jordan blocks up to similarity. This means that for an operator B on K , there is a bounded maximal abelian set of idempotents Q in {B} ′ and Q is unique up to similarity in {B} ′ . However, to represent certain operators on H , direct sums of Jordan blocks need to be replaced by direct integrals of strongly irreducible operators with regular Borel measures. In [14], we proved that an operator A on H is similar to a direct integral of strongly irreducible operators if and only if its commutant {A} ′ contains a bounded maximal abelian set of idempotents. Related concepts about direct integrals can be found in [3,4].
For an operator A on H , A direct integral decomposition of A is said to be a strongly irreducible decomposition (S. I. D.) of A if the integrand is strongly irreducible almost everywhere on the domain of integration. An S. I. D. of A is said to be unique up to similarity if for bounded maximal abelian sets of idempotents P and Q in {A} ′ , there is an invertible operator X in {A} ′ such that XPX −1 = Q.
In this paper, we study when A has unique S. I. D. up to similarity, for A similar to a direct integral of strongly irreducible operators.

Upper triangular representation and main theorems
If an operator A in L (H ) is similar to a direct integral of strongly irreducible operators, then there is an invertible operator X in L (H ) such that XAX −1 has an S. I. D. in the form Here µ is a regular Borel measure. Write Λ for n=∞ n=1 Λ n . The sets Λ ∞ and Λ n for n in N are bounded Borel and pairwise disjoint. For n in N and almost every λ in Λ n , the dimension of the fibre space H λ is n. For almost every λ in Λ ∞ , the dimension of the fibre space H λ is ∞. (For fibre space, see [1], §2.) Some Λ n s and Λ ∞ may be of measure zero. The partitioned measure space corresponding to the S. I. D. of XAX −1 is denoted by {Λ, µ, {Λ n } n=∞ n=1 }. For a nonzero normal operator N on H , the tensor product I ⊗ N does not have unique S. I. D. up to similarity, where I is the identity operator on H and dimH = ∞. If A is similar to a normal operator N , then the S. I. D. of A is unique up to similarity if and only if the multiplicity function m N for N is finite a. e. on σ(N ). Based on this, we can construct a non-normal operator which does not have unique S. I. D. up to similarity, if µ(Λ ∞ ) = 0 in (1). If µ(Λ ∞ ) = 0 then the S. I. D. of XAX −1 is of the form By ( [2], Corollary 2), there is a unitary operator U such that the equation holds a. e. on Λ and U (XAX −1 )U * (λ) is upper triangular in M n (C) for λ a. e. in Λ n . Write µ n for µ| Λn , 1 ≤ n < ∞. Without loss of generality, we assume that where φ n , φ n ij ∈ L ∞ (µ n ), M φn and M φ n ij are multiplication operators. The scalarvalued spectral measure for M φn is ν n ≡ µ n • φ −1 n . Let the set {Γ nm } m=∞ m=1 be a Borel partition of σ(M φn ) corresponding to the multiplicity function m φn for M φn on σ(M φn ), where m φn (λ) = m, ∀λ ∈ Γ nm . Write ν nm for ν n | Γnm , 1 ≤ m ≤ ∞. We find that the functions m φn play a significant role on studying the uniqueness of S. I. D. of A up to similarity. (Note that φ n ij does not stand for φ ij to the power of n here. The symbol n is only a superscript.) In the rest of this paper, we write the partitioned measure space corresponding to the S. I. D. of XAX −1 as {Λ, µ, {Λ n , m φn } ∞ n=1 }. The purpose of this paper is to give a sufficient condition such that the S. I. D. of A in (2) is unique up to similarity. Precisely we prove the following theorems.
Theorem 2.2. For a fixed n, assume that T ∈ L (H ) is a direct integral of upper triangular strongly irreducible operators and the corresponding measure space is {Λ n , µ n , {Λ n , m φn }} as in (3). If there is a unitary operator U such that both hold, where ψ n,ij m and z m are in L ∞ (ν nm ), and z m (t) = t, ∀t ∈ Γ nm . then the following statements are equivalent: (1) The bounded ν n -measurable multiplicity function m φn is simple on σ(M φn ).
(2) The S. I. D. of T is unique up to similarity.
The condition in this theorem is significant and reasonable. We show this in the proofs in §3. The following theorem is a generalized version of the above theorem.  (3). The set Λ n is of µ-measure 0 for all but finitely many n in N. If the spectral measures for {M φn } ∞ n=1 are mutually singular and there is a unitary operator U ∈ {T } ′ satisfying the condition in Theorem 2.2 w. r. t. φ n and φ n ij (i > j) on every Λ n , then the following statements are equivalent: (1) The bounded ν n -measurable multiplicity function m φn is simple on σ(M φn ) for n in N.
The rest of this paper is organized as follows. In Section 3, first we prove a special case of Theorem 3.3 in Lemma 3.2 and then we prove Theorem 3.3 in three lemmas. Corollary 3.7 is to characterize the K 0 group of the commutant of the operator T in Theorem 3.3. In Example 3.8, we construct an operator and compute the corresponding K 0 group. Finally, we prove Theorem 2.2 and Theorem 2.3. The operator T in Theorem 3.3 indicates why we add a condition about the unitary operator in Theorem 2.2.

Proofs
The following lemma reveals an important property which is applied in other lemmas in this paper. (1). Multiplication operators M φ and M φij are on L 2 (µ n ). Then the following upper triangular form where α = σ(T (λ)). This equation implies that x ij = 0 for i > j and α i−1,i x ii = x i−1,i−1 for i = 2, 3, . . . , n. Therefore the invertibility of X implies that α i−1,i = 0 holds for i = 2, 3, . . . , n.
On the other hand, if α i−1,i = 0 holds for i = 2, 3, . . . , n, then every operator X in M n (C) satisfying T (λ)X = XT (λ) can be expressed in the form If X is an idempotent, then it must be I or 0. Thus T (λ) is strongly irreducible.
Lemma 3.2. Suppose that an operator T is assumed as in (4), φ i,i+1 (λ) = 0 holds a. e. on Λ n for i = 1, 2, . . . , n − 1, and φ is one to one a. e. on Λ n . Then the S. I. D. of T is unique up to similarity.
Equivalently, we need to prove that for every Borel subset σ of σ(M φ ) and X ∈ {T } ′ , the projection E (n) (σ) reduces X.
If we write µ n1 for µ n | φ −1 (σ) and µ n2 for µ n | Λn\φ −1 (σ) , then we have The operators T and X can be expressed in the form The equation T 1 X 12 = X 12 T 2 and the fact that M 1 φ and M 2 φ have mutually singular scalar-valued spectral measures imply that X 12 = 0. In the same way we obtain X 21 = 0. Therefore X ∈ {M (n) φ } ′ . By Lemma 3.1, we compute the equation T X = XT and obtain that the operator X has the form where ψ, ψ ij ∈ L ∞ (µ n ). Hence every idempotent in {T } ′ is a spectral projection of M (n) φ . This means that in the commutant of T , there is one and only one bounded maximal abelian set of idempotents. We denote by P the set of projections in {T } ′ . This is the only maximal abelian set of idempotents in {T } ′ . The set P ⊕ · · · ⊕ P(m copies) is a bounded maximal abelian set of idempotents in M m ({T } ′ ). We prove Theorem 3.3 in three lemmas.
is an idempotent, then there is an invertible operator X ∈ M m ({T } ′ ) such that the operator XQX −1 belongs to the set P ⊕ · · · ⊕ P(m copies).
Proof. The idempotent Q is decomposable with respect to the diagonal algebra generated by the set P (m) ≡ {P ⊕ P ⊕ · · · ⊕ P (m copies) : P ∈ P}. The measure space is Λ n . Based on (5) in Lemma 3.2, the operator Q can be expressed in the form There is a unitary operator U 1 such that the operator Q 1 = U 1 QU * 1 is represented as a block upper triangular operator-valued matrix in the form for 1 ≤ i < j ≤ n. Notice that Q 1 11 is an idempotent. Next, we prove that there is an invertible operator X 2 in M m (L ∞ (µ n )) such that X 2 Q 1 11 X −1 2 is a projection in the form where S i is a Borel subset of Λ n for i = 1, 2, . . . , m.
For any positive integer k, there is a positive integer l k such that given any idempotent P in M m (C) with norm less than k there is an invertible operator X with norm less than l k such that XP X −1 is similar to the corresponding Jordan block. That is because any idempotent in M m (C) is unitarily equivalent to a block matrix in the form I R 0 0 and For the set defined in ( the set π 1 (E l k ) contains every idempotent whose norm is less than k. By ([2], Theorem 1), the Borel map φ l k : π 1 (E l k ) → π 3 (E l k ) is bounded. Therefore the equivalent class of φ l ⌈||Q 1 11 ||⌉ • Q 1 11 (·) is the X 2 we need in M m (L ∞ (µ n )).
The set S i is a Borel subset of Λ n for i = 1, 2, . . . , m, and equals the following projection First, multiply each entry in the lower triangular m × m matrix form of Q 3 i,i+1 by −1 and denote this new m × m matrix form by X 3 i,i+1 , for i = 1, 2, . . . , n − 1. In {U 1 T (m) U * 1 } ′ , we can construct an operator X 3 1 in the form The operator X 3 1 is invertible and σ(X 3 Repeat the above procedure. We construct invertible operators X 3 i one by one in {U 1 T (m) U * 1 } ′ , for i = 1, . . . , n − 1. After n − 1 steps, we obtain Q 4 . Denote by X 4 the product of X 3 i s. The equation such that XQX −1 is a projection in P ⊕ · · · ⊕ P(m copies).
Proof. By the above lemma, we know that for every Q in Q, there is an invertible operator X in M m ({T } ′ ) such that XQX −1 is a projection in P⊕· · ·⊕P(m copies). Thus we define a function The function r Q is in the equivalent class of certain simple function. To prove this lemma, we only need to show that there are m idempotents Q i in Q such that the equation r Qi (λ) = 1, for i = 1, . . . , m holds a. e. on Λ n and Q i Q j = 0, for i = j. We prove this in two steps.
Step 1, we prove that there is an idempotent Q ′ in Q such that the relation 0 < r Q ′ (λ) < m holds a. e. on Λ n .
If the relation {r Q (λ) : Q ∈ Q} = {0, m} holds a. e. on Λ n , then we can construct a strongly measurable operator-valued constant function Q ′ (·) satisfying the following properties: (1) Q ′ (·) is nontrivial a. e. on Λ n . This contradicts with the assumption that Q is a maximal abelian set of idempotents. Therefore, there are an idempotent Q ′ 1 in Q and a Borel subset Λ n1 of Λ n with nonzero measure such that the relation 0 < r Q ′ 1 (λ) < m holds a. e. on Λ n1 . Thus there are an idempotent Q ′ 2 in Q and a Borel subset Λ n2 of Λ n \Λ n1 with nonzero measure such that the relation 0 < r Q ′ 2 (λ) < m holds a. e. on Λ n2 . Carry out this procedure and we obtain a subset The idempotent Q ′ is what we want in step 1. Note that r Q ′ is in the equivalent class of a simple function. We can write Λ n in the form of a union of disjoint Borel subsets Λ ′ ni of Λ n such that the equation r Q ′ (λ) = i holds a. e. on Λ ′ ni . (Some Λ ′ ni s may be of measure zero.) Write ]QQ ′ : Q ∈ Q}, for i = 1, . . . , m − 1.
Step 2, we prove that for a fixed i larger than 1, if the set Λ ′ ni is not of measure zero, then there is an idempotent Q ′′ ∈ Q i such that the relation 0 < Q ′′ (λ) < i holds a. e. on Λ ′ ni .
Suppose that the relation {r Q (λ) : Q ∈ Q i } = {0, i} holds a. e. on Λ ′ ni . By the above lemma we know that there is an invertible operator X ∈ M m ({T } ′ ) such that the equation . . .
holds a. e. on Λ ′ ni . Therefore we can construct an idempotent Thus Q is not a maximal abelian set of idempotents. This is a contradiction. Therefore there are an idempotent Q ′′ 1 ∈ Q i and a Borel subset Λ ′ ni1 of Λ ′ ni with nonzero measure such that the relation 0 < r Q ′′ 1 (λ) < i holds a. e. on Λ ′ ni1 . Thus there are an idempotent Q ′′ 2 ∈ Q i and a Borel subset Λ ′ ni2 of Λ ′ ni \Λ ′ ni1 with nonzero measure such that the relation 0 < r Q ′′ 2 (λ) < i holds a. e. on Λ ′ ni2 . Carry out this procedure and we obtain a subset This idempotent Q ′′ is what we want in step 2.
After finite steps, we obtain that there is an idempotent Q in Q such that the equation r Q (λ) = 1 holds a. e. on Λ n .
Repeat the above procedure, we can find m idempotents Q i in Q such that the equation r Qi (λ) = 1, for i = 1, . . . , m holds a. e. on Λ n and Q i Q j = 0, for i = j. Thus we can obtain 2 m idempotents that we need. Proof. By the above lemma, we can find m idempotents Q i in Q such that the equation r Qi (λ) = 1, for i = 1, . . . , m holds a. e. on Λ n and Q i Q j = 0, for i = j.
is a projection in P ⊕ · · · ⊕ P(m copies). The invertible operator X 1 can be chosen such that X 1 Q 1 X −1 1 is in the form Thus there is an invertible operator are in the form By this procedure, we obtain is the invertible operator that we need.
With the above three lemmas, we finish the proof of Theorem 3.3.
Corollary 3.7. If an operator T is assumed as in Theorem 3.3, then the K 0 group of {T } ′ is isomorphic to the set We give an example to show that the K 0 group of the commutant of an operator T as in the above Corollary is isomorphic to the corresponding set as (7).
Denote by T the 2 × 2 operator-valued matrix in the form By Lemma 3.2, we know that every idempotent P in {T } ′ is of the form where S is a Borel subset of [0, 1]. By Theorem 3.3, we know that, for any positive integer m, in {T (m) } ′ every idempotent is similar to a projection in the form  } ′ such that XP X −1 is a diagonal projection. The operator X is what we need. Lemma 3.10. If an operator T ∈ L (H ) is assumed as in Lemma 3.2, then for every idempotent Q in {T (∞) } ′ , there is an invertible operator X in {T (∞) } ′ such that XQX −1 is in P ⊕ · · · ⊕ P ⊕ · · · (∞ copies).
By the proofs of Lemma 3.4 and Lemma 3.9, we obtain this lemma.
Proof of Theorem 2.2. By Theorem 3.3 and Corollary 3.7, we obtain (1) ⇒ (2) and (1) ⇒ (3). When the multiplicity function m φn for M φn takes ∞ on σ(M φn ), we can construct two bounded maximal abelian sets of idempotents in the commutant of T which are not similar to each other in {T } ′ . By Lemma 3.10, we know that if m φn takes ∞ on σ(M φn ), then the K 0 group of {T (λ)} ′ is 0 a. e. on Γ n∞ .
Proof of Theorem 2.3. Denote by T n the restriction of T acting on (L 2 (µ n )) (n) . The operator T can be expressed as The rest of the proof is essentially an application of Theorem 2.2.