91Ó°ÊÓ

Injective operators are fundamental in understanding linear transformations. An operator, say \( T \), is injective if it has the property that it maps distinct elements from its domain \( \mathscr{D}(T) \) to distinct elements in the codomain \( H \). This implies that if \( T(x_1) = T(x_2) \), then necessarily \( x_1 = x_2 \). In other words, injectivity means there are no two different input values that result in the same output. This unique mapping feature makes injective operators one-to-one mappings.

• An important consequence of **injective** self-adjoint operators is having a trivial kernel, i.e., \( \text{kernel}(T) = \{ 0 \} \).


• Injective transformation ensures that no information is lost when an element of \( \mathscr{D}(T) \) is mapped to \( H \).

Understanding injective operators provides a critical foundation for further explorations of operator characteristics, such as the range and image, leading into the concept of closures.

The concept of *closure of the image* is pivotal in linear algebra, especially concerning operators. When we refer to the closure of the image \( \overline{\mathscr{M}(T)} \), we're looking at the smallest closed set containing the image of the operator \( T \). In the context of the problem, where \( T \) is self-adjoint and injective, the closure of its image spans the entire space \( H \) (i.e., \( \overline{\mathscr{M}(T)} = H \)).

• This indicates that the image is dense in \( H \), meaning for every point in \( H \), there is a sequence of image points that converge to it.


• Self-adjointness ensures that this occurs when the operator's kernel is 0, confirming functionality across the entire space of \( H \).

Thus, when an injective operator is also self-adjoint, it naturally follows that its image closure fills up the whole space, leading us to its natural continuation with inverse operations.

Inverse operators allow us to reverse the transformation applied by the original operator. For an injective and surjective operator \( T \) (which is bijective), an inverse operator \( T^{-1} \) exists. This means that for every element \( y \) in \( H \), there is a unique \( x \) in \( \mathscr{D}(T) \) such that \( T(x) = y \).

• The existence of \( T^{-1} \) implies \( T \) is both injective and surjective, effectively mapping \( H \) onto itself.


• If the operator \( T \) is self-adjoint, its inverse \( T^{-1} \) is also self-adjoint. This property arises from interacting with the commutativity of adjoint operations: \( T^{-1} = (T^*)^{-1} = T^{-1}* \).

The implications of inverse self-adjoint operators ensure continued consistency of property throughout operations, a critical consideration in functional analysis.