A Weyl-von Neumann Theorem For N-tuples Of Commuting Normal Operators

Throughout this paper (?)(H) will always denote a complex separable infinite dimensional Banach(Hilbert) space. Denote by B((?))(B(H)) the set of bounded linear operators on (?)(H). C denotes the complex plane. Z denotes the integral set.Κ(H) denotes the ideal of compact operators on H.In the operator theory, the spectrum of an operator is very important quantity. The spectral theory of single operator has been extensively developed and many significant results have been obtained. The spectral theory of self-adjoint operator, as the generalization of unitarily equivalent of symmetric matrix, takes up an important position. Also, it is a valuable implement for the study of observational physical quantity in quantum statistical mechanics. For n-tuples of operators, one may consider the notion of joint spectrum. joint spectrum of an n-tuple of operators was first introduced by R. Arens and A. P. Calderon[21]. Later people studied this notion and introduced many similar versions of the notion of joint spectrum. Perhaps the most successful one is due to Taylor [1]. In addition, Dash's joint spectrum[20] is very important as it. Both of them are on the re-tuples of commuting operators.Let A = (A₁,…, A_n) be an n-tuple of operators on H, A is said to be commuting if A_iA_j = A_jA_i for 1≤i, j≤n.In 1972, R. Harte generalized the joint spectrum of n-tuples of operators to that n-tuples of elements in a complex linear algebra with identity.Definition: If A is a complex linear algebra with identity. Let a = (a₁,…,a_n) be an n-tuple of elements in A. Then the joint spectrumσ(a) is defined asσ(a) =σ^l(a)∪σ^r(a),where the left (right) joint spectrumσ^l(a)(σ^r(a)) is defined as the set of all points z = (z₁,…, z_n) in Cⁿ such that {a_j- z_j}_{1≤j≤n} generates a proper left (right) ideal in the algebra A.One may also pay attention to the Calkin algebra. LetÏ€be the canonicalhomomorphism from B(H) onto the (quotient) Calkin algebra A(H) := B(H)/Κ(H). For an n-tuple of operators A = (A₁,…, A_n), we writeÏ€(A) for the n-tuple (Ï€(A₁),…,Ï€(A_n)) in A(H). In [3], Dash introduced the concept of joint essential spectrum for a sequence of operators.Definition: Let A = (A₁…, A_n) be an n-tuple of essentially commuting operators . The joint essential spectrum of A, denoted byσ_e(A), is defined as the joint spectrumσ(Ï€(A)) ofÏ€(A), and speciallyσ_e^l(A) :=σ^l(Ï€(A)) (σ_e^r(A) :=σ^r(Ï€(A))) is called left (right) joint essential spectrum.In this paper we discuss Taylor joint essential spectrum Sp_e(A) and joint spectrum in Calkin algebra A(H).For a single operator, the essentially unitarily equivalent is well known. Given A₁,A₂∈B(H), A₁ and A₂ are called essentially unitarily equivalent if there exists a unitary operator U∈B(H) such that AU - BU∈Κ(H). Recall thatσ_e(A) =σ(Ï€(A)) is called the essential spectrum of A.It is well know theorem of WN that if A, B∈B(H) are self-adjoint. Then A and B are essentially unitarily equivalent if and only ifσ_e(A) =σ_e(B). Therefore the essential spectrum is a complete invariant of essentially unitarily equivalent.In [16], Y. S. Samoilenko introduced the notion of jointly unitarily equivalentof two families of self-adjoint operators. Let A = (A_i)_i=1^∞, B= (B_i)_i=1^∞be two families of self-adjoint operators, then A and B are called jointly unitarily equivalent if there exists a unitary operator U such that U^*A_iU = B_i for i≥1.Corresponding to the definition of essentially unitarily equivalent of operators,one would naturally give the following definition.Definition: Let A = (A₁,…, A_n) and B = (B₁,…, B_n) be two n-tuples of operators. We say A and B are jointly essentially unitarily equivalent if there exists a unitary operator U such that U^*A_iU - B_i∈K(H) for 1≤i≤n.In view of Theorem WN, it is natural to ask the question: if the joint essential spectrum is a complete invariant of jointly essentially unitarily equivalent?More precisely, if A and B are n-tuples of self-adjoint operators, then does Sp_e(A) = Sp_e(B) equal that they are jointly essentially unitarily equivalent?We notice that if A = (A₁,…, A_n) and B = (B₁,…, B_n) are n-tuples of self-adjoint operators. And Sp_e(A) -Sp_e(B), in virtue of the projection property, so we haveσ_e(A_j) =σ_e(B_j)(1≤j≤n). Then by Theorem WN, there exists a unitary operator U_j such that U_j^*A_jU_j- B_j is compact for 1≤j≤n. Can we guarantee that these U_j(1≤j≤n) are the same one?The purpose of this paper is to extend Theorem WN to the case of retuplesof commuting self-adjoint operators. But in general this is difficult. In this paper, for pairs of commuting self-adjoint operators and a special class of n-tuples of normal operators, we prove that joint essential spectrum is a complete invariant of jointly essentially unitarily equivalent and therefore generalizeTheorem WN. The following theorems are the main results of this paper.Main Theorem 1. Let A = (A₁.A₂), B = (B₁, B₂) be two pairs of commuting self-adjoint operators. Then Sp_e(A) = Sp_e(B) if and only if A and B are jointly essentially unitarily equivalent.Main Theorem 2. Let A = (A₁,…, A_n), B = (B₁,…, B_n) be two commutingn-tuples of self-adjoint operators. If{Ï€(A_j) : 3≤j≤n} (?) A"(Ï€(A₁)), where A"(Ï€(A₁)) denotes the double commutant ofÏ€(A₁), then Sp_e(A) = Sp_e(B) if and only if A and B are jointly essentially unitarily equivalent.Main Theorem 3. Let T = (T₁,…, T_n), W = (W₁,…, W_n) be two commutingn-tuples of normal operators. If {Ï€(T_j) : 2≤j≤n)(?)A"(Ï€(T₁ + T₁^*)), then Sp_e(T) = Sp_e(W) if and only if T and W are jointly essentially unitarily equivalent.