Operators Similar To The Restrictions On Their Invariant Subspaces

In this paper, let C be the complex field, D, T be the open unit disk and unit cricle on C respectively. X (H) is a complex separable Banach (Hilbert) space, and denote by B(X), the algebra of all bounded linear operators acting on X and K,(X), the ideal of compact operators in B(X), respectively. For T ∈ B(X), let A(T),A'(T) and A"(T) be the closure in the weak operator topology of the set of polynomials of T, the commutant of T and the double commutant of T, respectively. We denote the adjoint of T by T*, the conjugate of T by T".A long-standing problem in operator theory has been to determine whether every (bounded linear) operator T on a separable Hilbert space (or, more generally, a Banach space) must have a nontrivial invariant subspace. This is the well-known invariant subspace problem. This problem has been intensively studied, especially for Hilbert spaces, and positive results have been obtained for many classes of operators. For example: Beurling theorem, Burnside theorem, Lomonosov theorem, Riesz decomposition theorem and so on. On Banach spaces, there have been several extraordinary construction of counterexamples. V.Lomonosov has pointed out that the examples of P. Enflo and C. J. Read are not adjoints of any operators, so it is therefore conceivable that every operator which is an adjoint has nontrivial invariantsubspaces.We can consider that which kind of properties should T have, when T has a (or, hasn't any) nontrivial invariant subspace? In this paper, we will investigate crystals operators, which are similar to the restrictions on their invariant subspaces. Definition 0.1 Suppose that T E B(X),(1) Say that T a crystal (or say that T is crystal), if every restriction of T on its nonzero invariant subspaces is similar to T.(2) Say that T is crystal like, if for every nonzero M E LatT, there exists a positive number r such that T\m is similar to rT, denoted by T\m rT.(3) Say that a crystal T is rigid, if for every nonzero M E LatT, there exists an isometrcal invertible operator V : X —> M such that V^lTV = T\M, denoted by T\M T.(4) Say that T is rigid like, if for every nonzero M ￡ LatT, there exists a positive number r such that T\m — rT.For VA/ € LatT, define TM : XjM -> X/M,TM(x + M)= Tx + M, V(x + M) € X/M.Since M is an invariant subspace of T. the definition of TM is reasonable.(5) Say that T is bi-crystal, ifT\M TM T, MM € LatT \ {X, {0}}. From the above definitions, it is obvious that{rigid crystal}C{crystal}, {crystal}C{crystal like},{rigid crystal}C{rigid like}, {rigid like}C{crystal like}.But rigid like and crystal do not be contain each others. Since the transitivityof "^{, "}, then operators which are similar to crystals operators are crystalsoperators still.We divided the main content of this book into the following parts.(I). We investigate some elementary properties of crystals operators. We investigate firstly properties of the spctrum of T. Theorem 0.2 Let T be crystal like, then(1) ct(T) is connected, T's -point spectrum crp(T) = 0, Browder spectrum T = {0}. Particularly, ifT is crystal, then o^p(T') is open.(2) dim Ker {V - A)n = n, VA e 1.Theorem 0.3 Let T be crystal like, if 0 € cr(T)\(Xe(T), or av{V) = {0},then f] RanT — {0}. Here, A'(T) is commutative.71 = 1secondly, we investigate compact operator and rigid crystal. Theorem 0.4 Let T be crystal like. IfT is compact, then a{T) = {0}. // crp(T') ^ 0 and <7P(T') isn'i open, i/ien &ere is a positive integer number n such that Tn is compact .Let Lp(T) and J/p(T), 1 < p < oo, be the Hardy spaces, 7^ be the analytical Tocplitz operator with symbol cp € H°°(T), that isT^ : Hp(T) - iF(T), 7^/ = v/, V/ GTheorem 0.5 T2 € Z3(//P(T)) is rigid crystal. If if € //°°(D), t/ien T^ ?s rigid crystal <=> 7^ is a crystal operator <=$■ LatT^ = LatTz.(II). we investigated similarity of crystals operators.For T € B(H), we can consider its similar invariants by two means. One method is that establishing the Jordanian canonical form theorem on infinite dimensional spaces. It has been demonstrated that strongly irreducible (denoted by SI) operator is a rather suitable anologue of Jordanian blocks by the functional analysis seminar of JiLin University. The other is thatby weakening or strengthening the similarity. The most known results arc Fuglede-Putnam theorem, BDF theorem, similarity orbit theorem and so on. Definition 0.6 Call that A ￡ ES(X), if there exists a compact operator K and a subspace M ofcodimM = 1 such that PmTPm\m ^T + K, where PM is a projection from X onto M.Theorem 0.7 Let A, B € ES{X). Then A is similar to B in the Calkin algebra if and only if there exists a compact operator K such that A + K B. Theorem 0.8 Let A and B be crystal like. If op{A') and ap{B') are notempty sets, then A, B are similar m the Calkin algebra C{H) if and only ifthere exists a compact operator K such that A + K B.About irreducible and reflexive properties of crystals operators, we haveTheorem 0.9 Let T be crystal like, if ap(T') ^ 0 or A'{T) is commutative,then T is Banach irreducible.Definition 0.10 Call A € B{X) be affine inflation, if A A ? A.Theorem 0.11 Affine inflation is reflexive.Theorem 0.12 If T € B(X) is bi-crystal , then the following statementsare equivalent.(1) There exists a nontrivial subspace ofT.(2) T is Banach reducible.(3) T is reflexive.Theorem 0.13 If T is crystal, then LatT = LatA"(T). IfA'(T) is commutative, then LatT = LatA'(T).For V T G B(X), it is not a sufficient condition of LatT = Lat^'(T) that A'(T) is commutative.(Ill), we established the Jordanian canonical form theorem in a subset of B(H). In 1985, Jiang ZeJian gave an example, which showed that we can'testablish the Jordanian canonical form theory in the whole algebra B(H) as the same as on the finite dimension space. But we don't give up, because there are much significant work about some special subalgebras, which arc not about the whole B(H).Theorem 0.14 // T is similar to quasinorrnal operator, and has a (SI) decomposition, then T ?(c> + J\), where A C cr(T), cA E a(T)\J{Q}, J\ = \X\S, V X € A, S is the unilateral shift, Jo is the zero operator on the Hilbert space of one dimension.