Real and Complex Analysis (Rudin)

X n , the one-dimensional spaces spanned by any of these Xi are S-invariant, and
we obtain a very simple description of S if we take {Xl' ... ' Xn} as a basis of X.
We shall describe the invariant subspaces of the so-called "shift operator" S
on t Z. Here t Z is the space of all complex sequences
for which
(1)
Ilxll = LtlenlZf'Z < 00,
(2)
and S takes the element X e t Z given by (1) to
Sx = {O, eo, el' ez, ... }.
It is clear that S is a bounded linear operator on t Z and that IISII = 1.
A few S-invariant subspaces are immediately apparent: If lk is the set of all
X e t Z whose first k coordinates are 0, then lk is S-invariant.
To find others we make use of a Hilbert space isomorphism between t Z and
HZ which converts the shift operator S to a multiplication operator on HZ. The
point is that this multiplication operator is easier to analyze (because of the
. richer structure of HZ as a space of holomorphic functions) than is the case in the
original setting of the sequence space t Z •
We associate with each X e t Z , given by (1), the function
(3)
••• , IXk} C U, the space Y of allfe HZ such thatf(IXI) =
is S-invariant. If B is the finite Blaschke product with zeros at 1Xl>
... = f(lXk) = °
..., IXk' thenfe Yifand only ifPB e HZ. Thus Y = BHz.
(6)