ISSN:
1572-9176
Keywords:
Hilbert space
;
bounded linear operators
;
weakly periodic sequences
;
spectral representation
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Let X and Y be two Hilbert spaces, and $$\mathcal{L}(X,Y)$$ the space of bounded linear transformations from X into Y. Let {A η} ⊂ $$\mathcal{L}(X,Y)$$ be a weakly periodic sequence of period T. Spectral theory of weakly periodic sequences in a Hilbert space is studied by H. L. Hurd and V. Mandrekar (1991). In this work we proceed further to characterize {A n} by a positive measure μ and a number T of $$\mathcal{L}(X,X)$$ -valued functions a 0, . . . ,a T−1; in the spectral form $$A_n = \smallint _0^{2\pi } e^{ - i\lambda n} \Phi (d\lambda )Vn(\lambda )$$ , where $$Vn(\lambda ) = \sum\nolimits_{k = 0}^{T - 1} {e^{ - i\frac{{2\pi kn}}{T}} a_k } (\lambda )$$ and Φ is an $$\mathcal{L}(X,Y)$$ -valued Borel set function on [0, 2π) such that $$(\Phi (\Delta )x, \Phi (\Delta ')x')_Y = (x,x')_X \mu (\Delta \cap \Delta ').$$
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1022934611135
Permalink