Operators

What Is an Operator?

An operator is a mapping that acts on elements from a set and returns an element in a (possibly different) set. In mathematics, operators commonly act on vectors, matrices, or functions. Examples include adding numbers, multiplying by a scalar, applying a matrix to a vector, differentiating a function, or integrating a function.

• Unary operator: acts on one input (e.g., derivative D acting on a function f).
• Binary operator: combines two inputs (e.g., addition + on real numbers).
• General view: an operator T is a rule T: X → Y, often with X and Y vector spaces.

Core Components and Notation

• Domain and codomain: where inputs come from and outputs live; must always be specified.
• Identity I and zero 0 operators: do-nothing and send-everything-to-zero maps.
• Composition (S ∘ T) and powers (Tⁿ): build complex actions from simpler ones.
• Kernel (null space) and image (range): solutions to T(x) = 0 and all outputs T(x). In finite dimensions, rank–nullity links them.

Types of Operators

• Linear vs. nonlinear: linear operators satisfy T(ax + by) = aT(x) + bT(y).
• Bounded vs. unbounded (on normed spaces): bounded operators are continuous; many differential operators are unbounded and require careful domain specification.
• Differential and integral operators: Df = f′; (Kf)(x) = ∫ k(x,t)f(t) dt.
• Matrix/linear transformations: finite-dimensional linear operators represented by matrices.
• Multiplication and convolution operators on function spaces: (M_gf)(x) = g(x)f(x); (f ∗ g)(x) via integral sums.
• Projection, permutation, and shift operators: structure-preserving maps with special algebraic identities.
• Self-adjoint, unitary, normal (on inner-product spaces): classes with strong spectral properties.

Key Properties

• Linearity: simplifies analysis and allows matrix methods.
• Invertibility: T has an inverse T⁻¹ if it is bijective (and bounded inverse in normed spaces).
• Adjoint: T* satisfies ⟨Tx, y⟩ = ⟨x, T*y⟩; central in inner-product spaces.
• Commutator: [A, B] = AB − BA measures failure to commute; key in operator algebras and quantum theory.
• Spectrum and eigenvalues: generalizes roots of characteristic polynomials; eigenpairs (λ, v) satisfy T(v) = λv.

Working With Operators

• Matrix representation: choose a basis; linear operators correspond to matrices whose action is multiplication.
• Change of basis and similarity: S = P⁻¹TP; preserves eigenvalues and many invariants.
• Diagonalization and Jordan form: simplify powers and exponentials of operators.
• Functional calculus: define p(T) for polynomials p; extends to more general functions in advanced settings.

Examples

• Derivative on polynomials: D(xⁿ) = n xⁿ⁻¹. Kernel: constants; image: all polynomials of degree ≤ n−1.
• Shift on sequences: (Sx)_n = x_n+1. Not invertible on one-sided sequences; unitary on two-sided sequences with ℓ².
• Multiplication operator: (M_gf)(x) = g(x)f(x); spectrum relates to the essential range of g.
• Projection: P² = P; projects vectors onto a subspace, leaving it fixed.
• Integral operator: (Kf)(x) = ∫_a^b k(x, t)f(t) dt; compact under mild conditions on k.

Common Pitfalls

• Ignoring domains: many operators (especially unbounded ones like D on L²) require a carefully chosen domain.
• Assuming linearity: not all operators are linear; check definitions.
• Overloaded symbols: the same symbol may denote different operators depending on context; definitions must be explicit.

Connections and Applications

• Linear algebra: eigenvalues/eigenvectors, diagonalization, matrix decompositions.
• Calculus and differential equations: differential operators model rates of change and dynamics.
• Fourier analysis and signal processing: convolution, filters, and transform operators.
• Functional analysis: bounded/unbounded operators on infinite-dimensional spaces; spectral theory.
• Optimization and probability: gradient and Hessian operators; expectation as a linear operator.

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