shiftoperátor

The shiftoperátor, or shift operator, is a linear operator that maps a sequence or function to a shifted version of itself. It is a fundamental object in mathematics and engineering, used to model delays, time shifts, and recurrences.

In discrete time, the basic forms are the unilateral (forward) shift and the bilateral (two-sided) shift. On the sequence space ell^2(N) of square-summable sequences, the unilateral shift S is defined by S(x0, x1, x2, ...) = (0, x0, x1, x2, ...). This operator preserves norm (it is an isometry) but is not surjective. Its adjoint S* is the backward shift, given by S*(x0, x1, x2, ...) = (x1, x2, ...). The unilateral shift has spectrum equal to the closed unit disk, and it serves as a canonical example in operator theory, illustrating pure isometries and the Wold decomposition.

On the bilateral sequence space ell^2(Z), the shift T is defined by T(xn) = (x_{n-1}). This operator is unitary, its inverse is the backward shift, and its spectrum is the unit circle. The bilateral shift models a two-sided time shift and contrasts with the unilateral case in its invertibility and spectral properties.

In the context of signals and systems, a time shift x(t − t0) corresponds to a modulation by e^(-iωt0) in the frequency domain. Shifts are fundamental in linear time-invariant systems and in the design of delay elements, shift registers, and digital filtering.

Overall, shift operators provide a simple yet rich framework for studying dynamics, spectral theory, and applications involving delays and time shifts.