Small Compact Perturbation Of Operators

In this paper, C is the complex plane and H denotes a complex separable Hilbert space. B{H) denotes the algebra of all bounded linear operators acting on H, K(H)denotes the ideal of all compact operators on H and LatT denotes the lattice of invariant subspaces of T∈B(H).F. Gilfeather introduced the notion of strongly irreducibility of operator in 1972. An operator T is called a strongly irreducible operator if T is not similar to any reducible operator. In 1979, professor Jiang Zejian proposed the notions of Banach reducible operator and Banach irreducible operator. An operator T is called a Banach reducible operator if there exist two nontrivial invariant subspaces M and N of T such that M∩N = {0} and M + N = H; otherwise , T is called Banach irreducible. It is an easy exercise to prove for an operator T that T is strongly irreducible if and only if T is Banach irreducible.The Jordan canonical form of a matrix plays an important role in matrix theory. And if dimH <∞, then an operator T∈B(H) is Banach irreducible if and only if there is only one Jordan block in the Jordan canonical form of T. Since Banach irreducible operators, or briefly, BIR operators , and Jordan blocks on a finite dimensional space have many features in common, Jiang Ze-jian considers BIR operators as the approximate replacement of Jordan blocks on infinite dimensional Hilbert space and hopes that a theorem similar to the Jordan canonical form theorem in matrix theory can be set up with this replacement. Several problems concerning with BIR operators were studied at Jilin University and finally an approximate decomposition theorem of operator structure has been set up on separable infinite dimensional space and also in nest algebra(see [9],[21],[30],[34],[35],[36]).In the approximation associated with BIR operators, it is common to consruct various BIR operators with given spectral picture. The constructing methods are very technical. Some important results in non-commutative operator theory, such as BDF theorem and similarity orbit theorem, are also very useful. In particular, we note that Cowen-Douglas operators and B_∞(Ω) play a important role in the approximation of BIR operators(see [23],[30],[31],[37] ).In the approximation of operators, people always hope to use a special class of operators to approximate general operators in certain way. What Question H concern is that if BIR operators(of course spectrum-connected) could approximate operators with connected spectrum, furthermore, this approximation is " small compact " approximation. In [23], what professor Ji Youqing obtained is that the BIR quasitriangular operators could " small compact " approximate an arbirarily given quasitriangular operator.Maybe it is very difficult to consider " small compact " approximation in a smaller class of operators. Because the small compact perturbation of a given operator is required to be in (BIR) as well as in the class.In [30], Ji Youqing and Jiang Chunlan proved the following theorem.Theorem. n∈N,T∈B(H) is a B_n(Ω) operator, then for anyε> 0, there exist a K∈K(H) such that ||K|| <ε,T + K∈B_n(Ω)∩(BIR).This theorem indicates that B_n(Ω)(n∈N) operators which are Banach irreducible could approximate general B_n(Ω) operators.Thus, it is natural to ask wether B_∞(Ω) operators which are Banach irreducible could approximate general B_∞(Ω) operators?For T∈B_∞(Ω),Ωis contained inσ_e(T). This is quite different from Cowen-Douglas operators. This difference is essentially important. It makes it imposible to using the tecnique in [23] to answer this question.Thus, maybe it is qutie difficult to answer the question. We consider another question related to the above question. That is, if the direct sum of a B_∞(Ω) operator which is BIR and certain operator could have small compact perturbation which is also a BIR B_∞(Ω) operator? Maybe it is useful to answer the above question. In this paper, we prove the following two theorems.Main Theorem 1: Let N∈B(H₁),B∈B(H₂), B∈B_∞(Ω)∩(BIR), N is a normal operator, andσ(N) (?) (?)Ω,σ_P(B)∩(?)Ω=(?), then for anyε> 0, there exists a K∈K(H₁(?)H₂), such that ||K|| <ε,(N(?)B) + K∈B_∞(Ω)∩(BIR).Main Theorem 2 : Let A∈B(H₁),T∈B(H₂), T∈B_∞(Ω)∩(BIR),A is biquasitriangular,σ(A),σ_w(A) are both connected, andΓ(?)σ_w(A)∩(?)Ωis nonempty complete,σ_p(T)∩(?)Ω= (?) =σ(A)∩Ω, then for anyε> 0, there exists a K∈K(H₁(?) H₂), such that ||K|| <ε, (A (?) T) + K∈B_∞(Ω)∩(BIR).