Introduction to Finite Frame Theory - Frame Research Center

8 Peter G. Casazza, Gitta Kutyniok, and Friedrich Philipp
(b) The operator T is called injective (or one-to-one), if kerT = {0}, and surjective
(or onto), if ranT = H K . It is called bijective (or invertible), if T is both injective
and surjective.
(c) The adjoint operator T ∗ : H K → H N is defined by
(d) The norm of T is defined by
〈T x, ˜x〉 = 〈x,T ∗ ˜x〉 for all x ∈ H N and ˜x ∈ H K .
‖T ‖ := sup{‖T x‖ : ‖x‖ = 1}.
The next result states several relations between these notions.
Proposition 3. (i) Let T : H N → H K be a linear operator. Then
dimH N = N = dimkerT + rankT.
Moreover, if T is injective, then T ∗ T is also injective.
(ii) Let T : H N → H N be a linear operator. Then T is injective if and only if it is
surjective. Moreover , kerT = (ranT ∗ ) ⊥ , and hence
H N = kerT ⊕ ranT ∗ .
If T : H N → H N is an injective operator, then T is obviously invertible. If an
operator T : H N → H K is not injective, we can make T injective by restricting it to
(kerT ) ⊥ . However, T | (kerT ) ⊥ might still not be invertible, since it does not need to
be surjective. This can be ensured by considering the operator T : (kerT ) ⊥ → ranT ,
which is now invertible.
The Moore-Penrose inverse of an injective operator provides a one-sided inverse
for the operator.
Definition 6. Let T : H N → H K be an injective, linear operator. The Moore-
Penrose inverse of T , T † , is defined by
T † = (T ∗ T ) −1 T ∗ .
It is immediate to prove invertibility from the left as stated in the following result.
Proposition 4. If T : H N → H K is an injective, linear operator, then T † T = Id.
Thus, T † plays the role of the inverse on ranT – not on all of H K . It projects a
vector from H K onto ranT and then inverts the operator on this subspace.
A more general notion of this inverse is called the pseudo-inverse, which can be
applied to a non-injective operator. It, in fact, adds one more step to the action of
T † by first restricting to (kerT ) ⊥ to enforce injectivity of the operator followed
by application of the Moore-Penrose inverse of this new operator. This pseudoinverse
can be derived from the singular value decomposition. Recalling that by
fixing orthonormal bases of the domain and range of a linear operator we derive an