Understanding the Forward-Shift and Backward-Shift Operators in Infinite-Dimensional Hilbert Space
The concept of a Hilbert space is fundamental in functional analysis, providing a framework for studying vectors and operators in infinite-dimensional spaces. One intriguing example involves the Hilbert space (ell^2), which is the space of all complex or real sequences whose squares sum to infinity. This article delves into the intricacies of the forward-shift operator and its relationship with the backward-shift operator, illustrating how these operators behave within this infinite-dimensional space.
Introduction to (ell^2)
The (ell^2) space consists of all infinite sequences of real numbers that are square-summable, meaning that the sum of the squares of the sequence elements converges to a finite value. In mathematical terms, a sequence (x (x_1, x_2, x_3, ldots)) is in (ell^2) if and only if
(sum_{i1}^{infty} x_i^2
This space is equipped with an inner product and a norm, making it a Hilbert space. The inner product is defined as:
(langle x, y rangle sum_{i1}^{infty} x_i y_i)
The norm (or length) of a vector (x) in (ell^2) is then:
(|x| sqrt{langle x, x rangle} left(sum_{i1}^{infty} x_i^2right)^{1/2})
Since the space is complete with respect to this norm, (ell^2) is a Hilbert space.
The Forward-Shift Operator
The forward-shift operator S is a fundamental operator in (ell^2) space. Given a sequence (x (x_1, x_2, x_3, ldots)), the operator S shifts the elements one position to the right, discarding the first element:
Sx ((x_2, x_3, x_4, ldots))
This operator is continuous by construction, meaning that it preserves the convergence of sequences within (ell^2), and it is a linear operator.
The action of the forward-shift operator on the inner product of a sequence (x) and an arbitrary sequence (y) in (ell^2) is given by:
(langle Sx, y rangle sum_{i1}^{infty} x_i y_{i 1})
This can be re-written to show that we need a sequence (y) such that:
(langle x, S^*y rangle)
The Backward-Shift Operator
The backward-shift operator (S^*) is the adjoint of the forward-shift operator S. It is defined as shifting the elements one position to the left and filling the first position with 0. That is:
S^*y ((0, y_1, y_2, y_3, ldots))
The important point is that (S^*) is the adjoint of S if and only if:
(langle Sx, y rangle langle x, S^*y rangle)
By substituting the definitions of S and (S^*), we find that:
(langle Sx, y rangle sum_{i1}^{infty} x_i y_{i 1})
(langle x, S^*y rangle sum_{i1}^{infty} x_i (0, y_1, y_2, y_3, ldots)_{i 1})
(langle x, S^*y rangle sum_{i2}^{infty} x_i y_{i-1} sum_{i1}^{infty} x_i y_{i 1})
This shows that (S^*) is indeed the adjoint of S because it satisfies the condition of the inner product property.
Properties of (S^*S) and (S S^*)
Consider the compositions of the operators. The operator (S^*S) is given by:
(S^*Sx) ((x_2, x_3, x_4, ldots))
(langle S^*Sx, y rangle sum_{i1}^{infty} x_i y_{i 1})
In this case, the composition (S^*S) acts as the identity operator on (ell^2) sequences, meaning that (S^*S)
( text{id})
On the other hand, the composition (S S^*) is:
(S S^*x) (x) except that the first element of (x) is lost
(langle S S^*x, y rangle sum_{i1}^{infty} x_i y_i)
The composition (S S^*) is not the identity because the first element of the sequence is lost during the transformation:
(S S^*) does not preserve the norm of a sequence)
( eq text{id})
This property highlights the non-invertibility of the backward-shift operator (S^*) as it does not provide a complete mapping of sequences back to their original form.
Conclusion
The forward-shift and backward-shift operators in (ell^2) illustrate the intricacies of operators in infinite-dimensional Hilbert spaces. The forward-shift (S) and its adjoint, the backward-shift (S^*), are fundamental in understanding the structure and properties of this space. Despite (S^*S) being the identity operator, (S S^*) is not, demonstrating the non-invertibility of the backward-shift operator. This non-invertibility is a critical characteristic that sets the backward-shift apart from the forward-shift, emphasizing the importance of these operators in the study of Hilbert spaces.