Unicellular Operators And Attainable Nests

Davidson has raised an open question in 1988: Is every maximal nest equal to the invariant subspace lattice of a unicellular operator? This article is a survey article about the above problem, It introduces the introduction and background of the question and a number of meaningful results so far.We say an abstract lattices L is attainable if there is an operator A on a separable, infinite-dimensional Hilbert space such that the lattice of invariant subspaces(Lat A) is order isomorphic to L.The problem about attainable lattices is an important question related to the invariant subspace problem on a separable, infinite-dimensional Hilbert space. Actually, We can describe the well known invariant subspaces problem in the language of lattices. That is if a lattices of order type 2 can be attainable . It is an intuitive mind to begin with totally ordered lattices when we study the problem about attainable lattices. We call an operator A is unicellular if its lattice of invariant subspaces is totally ordered. The nest on a separable Hilbert space is a totally ordered subset of all closed subspaces. The abstract lattice of order isomorphic to the nest must be attainable if the nest is an operators' lattice of invariant subspaces. So we can first consider the problem about attainable nests which is the central problem when we consider the problem about totally ordered abstract attainable lattices.Nearly 50 years, It is in this regard some progress has been made. Although the problem has not been fully resolved, Mathematicians also have made some very significant results on this issue.(1) Every continuous nest is attainable.Donoghue showed that the Volterra operator on L²(0,1) has only the obvious invariant spaces in 1957, Let M_t=(f∈L²(0,1):f = 0 a.e. on [0,t]},Lat V = {M_t:t∈[0,1]}.Thus the Volterra nest is attainable.By the Similarity Theorem which is found by Larson, Every continuous nest is attainable as the lattice of an operator similar to Volterra operator.(2) The situation of atomic nests.Donoghue showed that certain weighted unilateral shifts have the invariant space lattice of order type w+1 in 1957, The nest of order type w+1 is attainable.A unicellular bilateral weighted shift with lattice order isomorphic to 1+Z+1 has been constructed by Domar in 1981,The nest of order type 1+Z+1 is attainable.By a unicellular operator which Harrison and Longstaff constructed in 1980, The nest of order type w+w+1 is attainable.Barr(?)a and Davidson simplify and generalize the method which was found by Harrison and Longstaff. Let B is acting on l^p (0* have cyclic vectors, ||Bⁿ||= 0((n!)^-d) for some d > 0. They showed that the nest of order typeα+Lat B+β^* is attainable for any countable ordinalsαandβ(they may be finite or zero, also). Since unicellular operators always have cyclic vectors , These hypotheses are readily seen to include most of the known examples of attainable atomic nests. It is worth mentioning that although this conclusion which every atomic nest is attainable does not prove out, There is no (maximal) nest on separable Hilbert space which is known to be unattainable. It seems plausible that all are attainable which was guessed by Barr(?)a and Davidson.(3)The situation of nests which is neither atomic nor continuous.Rosenthal gives a theorem about attainable ordinal sums which can piece togetherknown attainable lattices to produce new ones. He proved that a nest of order type [0,1]+n is attainable in 1970.Barr(?)a proved that certain integral operators V_μare unicellular operators and showed a nest of order type [0, 1]∪[2, 3]∪...∪[2n, 2n + 1] and ordinal sum of a finite number of [0, 1] + n are attainable.Although the mathematicians has made a lot of progress, It is very difficult to completely solve this problem as it is very difficult to completely solve the invariant subspace problem.Hou Jin Chuan from China proved that if L is an attainable totally ordered abstract lattice, then L+1 is attainable. On this basis, He raised a guess: if L is an attainable totally ordered abstract lattice, then L+w+1 is attainable. It will giveus a lot of help if the answer is yes. Actually, There is a more general question which can be made: If L₁¹+L₂ is attainable when L₁ and L₂ are attainable totally ordered abstract lattices ?...