operatorii

Operatorii (operatorii) are mathematical objects that map elements from one space to another, often preserving structure. In linear algebra, an operator on a vector space V over a field F is a function T: V → V that satisfies linearity: T(av + bw) = aT(v) + bT(w). When the map goes from a space to itself, it is called an endomorphism; if it targets a different space, it is a linear transformation. The set of all linear operators End(V) forms a vector space and, with an appropriate multiplication, an algebra.

In analysis, many operators are defined by rules such as differentiation or integration. Differential operators take the form D = d/dx, or more generally L = p(x)d/dx + q(x). Integral operators act by (Tf)(x) = ∫ K(x,y)f(y) dy and play a central role in solving integral equations. In normed or Hilbert spaces, an operator is bounded if it maps bounded sets to bounded sets, which implies continuity.

Key concepts include the domain and codomain of T, the kernel (solutions to T(v) = 0) and the image, rank, nullity, and invertibility. Adjoint operators T* arise from inner products, and special classes such as self-adjoint, unitary, and normal operators have important spectral properties. The spectrum of an operator generalizes eigenvalues and governs stability and dynamics in many contexts.

Representations by matrices connect operators to finite-dimensional linear algebra. Extensions include operator algebras, such as C*-algebras, generated by sets of operators on a Hilbert space. Applications span differential equations, quantum mechanics, signal processing, and numerical analysis.