Structure of Iso-Symmetric Operators

For a Hilbert space operator T∈B(H), let LT and RT∈B(B(H)) denote, respectively, the operators of left multiplication and right multiplication by T. For positive integers m and n, let ▵T∗,Tm(I)=(LT∗RT−I)m(I) and δT∗,Tn(I)=(LT∗−RT)m(I). The operator T is said to be (m,n)-isosymmetric if ▵T∗,TmδT∗,Tn(I)=0. Power bounded (m,n)-isosymmetric operators T∈B(H) have an upper triangular matrix representation T=T1T30T2∈B(H1⊕H2) such that T1∈B(H1) is a C0.-operator which satisfies δT1∗,T1n(I|H1)=0 and T2∈B(H2) is a C1.-operator which satisfies AT2=(Vu⊕Vb)|H2A, A=limt→∞T2∗tT2t, Vu is a unitary and Vb is a bilateral shift. If, in particular, T is cohyponormal, then T is the direct sum of a unitary with a C00-contraction.

REVERSED WAVELET FUNCTIONS AND SUBSPACES

Let the operators D and T be the dilation-by-2 and translation-by-1 on [Formula: see text], which are both bilateral shifts of infinite multiplicity. If ψ(·) in [Formula: see text] is a wavelet, then {DmTnψ(·)}(m,n)∈ℤ2 is an orthonormal basis for the Hilbert space [Formula: see text] but the reversed set {TnDmψ(·)}(n,m)∈ℤ2 is not. In this paper we investigate the role of the reversed functions TnDmψ(·) in wavelet theory. As a consequence, we exhibit an orthogonal decomposition of [Formula: see text] into T-reducing subspaces upon which part of the bilateral shift T consists of a countably infinite direct sum of bilateral shifts of multiplicity one, which mirrors a well-known decomposition of the bilateral shift D.

Chaos: Generating Complexity from Simplicity

The most commonly used mapping to illustrate the phenomenon of chaos is the map x → 2x mod (1). This map is known as the 'unilateral shift' because, in the binary number system this map shifts all digits to the left by one decimal place, and truncates the integer. The second most commonly used paradigm of chaos is the Smale horseshoe whose complexity is essentially the bilateral shift obtained when we simply shift without truncation in some symbol system. Neither of these paradigms fully explains chaos since shifts cannot generate complex orbits from simple (rational) initial conditions. How chaos generates complexity from simplicity is an essential part that needs explanation. Providing this explanation is the objective of this paper.