To generalize solutions involving constants or additions, use axioms A(c*x) = c * A(x) and A(x + y) = A(x) + A(y) {continuous additive operator}.
Linear bounded Hermitean operators can be operators {Einzeltransformation} E such that E-E- = E-, E+E+ = E+, and I = E- + E+. Einzeltransformations are commutative and commute with any operator that commutes with Hermitean operator. Hermitean times E- is greater than zero, and Hermitean times E+ is greater than zero.
In equation A(x) = x, operation or transformation A leaves function or vector constant {fixed-point analysis}. If operator is continuous and operates on n-dimensional spheres, function has at least one fixed point {Brouwer fixed-point theorem}.
Subsets can project {projection operator, set} onto element sets.
Operators {adjoint operator} {transposed operator} can find function scalar products, which are linear transformations from one function to another: (A(f1), f2). Adjoint operators can have inverses {self-adjointness}: (A(f1), f2) = (f1, A(f2)). The situation is analogous to the symmetric-integral-equation kernel. (T*f, g) = (f, T*g) and ||T*|| = ||T||, where T* is matrix-T transpose.
Adjoint-operator theory can apply to operators {Riesz operator} with form I - lambda * V, where lambda is parameter, I is identity operator, and V is complex continuous L^2-space operator.
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Date Modified: 2022.0225