We study rank-one perturbations of diagonal Hilbert space operators mainly from the standpoint of invariant subspace problem. In addition to proving some general properties of these operators, we identify the normal operators and contractions in this class. We show that two well known results about the eigenvalues of rank-one perturbations and one-codimension compressions of self-adjoint compact operators are equivalent. Sufficient conditions are given for existence of nontrivial invariant subspaces for this class of operators. Throughout the paper we shall suppose that u and v are nonzero vectors in H and their expansions with respect to the (ordered, orthonormal) basis {e n } are u = ∞ n=1 α n e n , v = ∞ n=1 β n e n. (2) Note that up to unitary equivalence, D 0 consists exactly of all sums N + R, where N is a normal operator whose eigenvectors span H and R is an operator of rank one. Note also that the inclusion D ⊂ N is a strict one. One way to see this is to make use of Kato and Rosenblum's result (cf. [20]) stating that the absolutely continuous parts of a selfadjoint operator and its selfadjoint trace class perturbation are unitarily equivalent. Observe that the expression for T in (1) is not necessarily unique. If we restrict our study, though, to the class D 1 of those operators in D 0 which admit a representation as in (1) with u and v having nonzero components α n and β n for all n ∈ IN, we have uniqueness in the following sense. PROPOSITION 1.1. If T ∈ D 1 then the representation (1) for T is unique in the sense that if T = Diag({λ n }) + (u ⊗ v) = Diag({λ n }) + (u ⊗ v), then Diag({λ n }) = Diag({λ n }) and (u ⊗ v) = (u ⊗ v). PROOF. We may assume T = Diag({λ n })+(u⊗v) = Diag({λ n })+(u ⊗v) where all the Fourier coefficients of u and v in (2) are not zero. This means that Diag({λ n }) − Diag({λ n }) = Diag({λ n − λ n }) = (u ⊗ v) − (u ⊗ v) has rank at most two. Thus, there exist different positive integers n 1 , n 2 such that λ k = λ k for all k ∈ IN \ {n 1 , n 2 }. Moreover the range of S = Diag({λ n − λ n }) is contained in ∨{e n 1 , e n 2 }, and so we may have three
It is proved that associated with every wavelet set is a closely related "regularized" wavelet set which has very nice properties. Then it is shown that for many (and perhaps all)
Simultaneous tiling for several different translational sets has been studied rather extensively, particularly in connection with the Steinhaus problem. The study of orthonormal wavelets in recent years, particularly for arbitrary dilation matrices, has led to the study of multiplicative tilings by the powers of a matrix. In this paper we consider the following simultaneous tiling problem: Given a lattice in L ∈ R d and a matrix A ∈ GL(d, R), does there exist a measurable set T such that both {T + α : α ∈ L} and {A n T : n ∈ Z} are tilings of R d ? This problem comes directly from the study of wavelets and wavelet sets. Such a T is known to exist if A is expanding. When A is not expanding the problem becomes much more subtle. Speegle [24] exhibited examples in which such a T exists for some L and nonexpanding A in R 2 . In this paper we give a complete solution to this problem in R 2 .Recently attentions have been given to multiplicative tilings. In a multiplicative tiling there is a finite set of prototiles {T 1 , T 2 , . . . , T m } and sets of nonsingular d × d matricesHere we define a partition in the most general sense, namely the sets are disjoint in Lebesgue measure and their union is R d up to a measure zero set, see e.g.
In this paper, we deal with the problem of bisecting binomial coefficients. We find many (previously unknown) infinite classes of integers which admit nontrivial bisections, and a class with only trivial bisections. As a byproduct of this last construction, we show conjectures Q2 and Q4 of Cusick and Li [7]. We next find several bounds for the number of nontrivial bisections and further compute (using a supercomputer) the exact number of such bisections for n ≤ 51.
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