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Operator Theory (Definition)

The study of linear operators on function spaces, particularly Hilbert or Banach spaces.
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The statement of the theorem

Let XX and YY be complex Banach spaces. An operator TT is a bounded linear map T:X→YT: X \to Y. The theory focuses on the algebra B(X,Y)\mathcal{B}(X, Y) of such operators. For the case X=YX=Y, we denote B(X)=B(X,X)\mathcal{B}(X) = \mathcal{B}(X, X).\n\nWe define the spectrum of TT, denoted σ(T)\sigma(T), as the set of complex numbers λ\lambda for which the operator (T−λI)(T - \lambda I) is not invertible in B(X)\mathcal{B}(X), i.e., σ(T)={λ∈C∣(T−λI)−1∉B(X)}\sigma(T) = \{ \lambda \in \mathbb{C} \mid (T - \lambda I)^{-1} \notin \mathcal{B}(X) \}.\n\nFurthermore, the theory often involves the adjoint operator. If XX and YY are Hilbert spaces, the adjoint T∗:Y→XT^*: Y \to X is defined uniquely by the relation ⟨Tx,y⟩Y=⟨x,T∗y⟩X\langle Tx, y \rangle_Y = \langle x, T^*y \rangle_X for all x∈X,y∈Yx \in X, y \in Y. The study of the spectral radius, r(T)=sup⁡n=0∞∥Tn∥1/nr(T) = \sup_{n=0}^{\infty} \|T^n\|^{1/n}, and the relationship between σ(T)\sigma(T) and r(T)r(T) (specifically, r(T)=sup⁡λ∈σ(T)∣λ∣r(T) = \sup_{\lambda \in \sigma(T)} |\lambda|) constitutes the core mathematical framework.