🧐Functional Analysis Unit 2 – Linear Operators in Normed Spaces

Key Concepts and Definitions

• Linear operators map elements from one vector space to another while preserving linear combinations
• Normed spaces consist of a vector space equipped with a norm that measures the "size" of vectors
• Domain and codomain refer to the input and output spaces of a linear operator, respectively
• Null space (kernel) contains all vectors mapped to the zero vector by the operator
  • Dimension of the null space is called the nullity of the operator
• Range (image) is the set of all vectors in the codomain that are outputs of the operator
  • Dimension of the range is called the rank of the operator

Properties of Linear Operators

• Linearity ensures that the operator preserves vector addition and scalar multiplication
  • $T(u + v) = T(u) + T(v)$ for all vectors $u$ and $v$ in the domain
  • $T(αu) = αT(u)$ for all scalars $α$ and vectors $u$ in the domain
• Injectivity (one-to-one) means distinct inputs map to distinct outputs
• Surjectivity (onto) means every vector in the codomain is an output of the operator
• Bijectivity combines injectivity and surjectivity, implying a one-to-one correspondence between domain and codomain
• Invertibility allows for the existence of an inverse operator that "undoes" the original operator
  • Invertible operators are always bijective

Bounded Linear Operators

• Bounded linear operators have a finite "amplification factor" for all input vectors
  • There exists a constant $M \geq 0$ such that $\|T(u)\| \leq M\|u\|$ for all $u$ in the domain
• Boundedness is a stronger condition than continuity for linear operators
  • All bounded linear operators are continuous, but not all continuous linear operators are bounded
• Bounded linear operators form a vector space, allowing for addition and scalar multiplication of operators
• Composition of bounded linear operators is also a bounded linear operator
• Inverse of a bounded linear operator (if it exists) is also bounded

Operator Norms

• Operator norms quantify the "size" or "amplification factor" of a linear operator
  • Defined as $\|T\| = \sup\{\|T(u)\| : \|u\| = 1\}$, where $u$ is in the domain
• Operator norms satisfy the properties of a norm on the space of bounded linear operators
  • Non-negativity: $\|T\| \geq 0$, with equality if and only if $T = 0$
  • Homogeneity: $\|αT\| = |α|\|T\|$ for all scalars $α$
  • Triangle inequality: $\|T + S\| \leq \|T\| + \|S\|$ for all bounded linear operators $T$ and $S$
• Commonly used operator norms include the operator norm induced by the $\ell^p$ norms on the domain and codomain spaces

Examples and Applications

• Differentiation operator maps functions to their derivatives in function spaces (Sobolev spaces)
• Integration operator maps functions to their integrals (Lebesgue spaces)
• Fourier transform is a linear operator that maps functions to their frequency representations
  • Plays a crucial role in signal processing and quantum mechanics
• Rotation matrices are linear operators that rotate vectors in Euclidean spaces
• Projection operators map vectors onto subspaces (orthogonal projections in Hilbert spaces)
• Compact operators generalize the notion of matrices to infinite-dimensional spaces
  • Integral operators and Hilbert-Schmidt operators are examples of compact operators

Continuity and Boundedness

• Continuity of linear operators is equivalent to boundedness in normed spaces
  • A linear operator $T$ is continuous if and only if it is bounded
• Continuous linear operators map convergent sequences to convergent sequences
  • If $u_n \to u$ in the domain, then $T(u_n) \to T(u)$ in the codomain
• Uniform boundedness principle states that a family of pointwise bounded linear operators is uniformly bounded
• Open mapping theorem asserts that a surjective bounded linear operator maps open sets to open sets
• Closed graph theorem characterizes continuous linear operators in terms of their graphs

Operator Algebras

• Operator algebras study the algebraic and topological properties of collections of linear operators
• Banach algebras are normed algebras that are complete as metric spaces
  • Space of bounded linear operators on a Banach space forms a Banach algebra
• C*-algebras are Banach algebras equipped with an involution (adjoint operation) satisfying certain axioms
  • Provide a framework for studying quantum mechanics and non-commutative geometry
• Von Neumann algebras (W*-algebras) are C*-algebras that are closed in the weak operator topology
  • Play a central role in the mathematical formulation of quantum mechanics

Advanced Topics and Extensions

• Spectral theory studies the eigenvalues and eigenvectors of linear operators
  • Spectrum of an operator generalizes the notion of eigenvalues to infinite-dimensional spaces
• Functional calculus allows for the definition of functions of operators
  • Holomorphic functional calculus extends this notion to analytic functions
• Fredholm theory deals with the solvability of linear equations involving compact perturbations of the identity
  • Fredholm alternative characterizes the existence and uniqueness of solutions
• Toeplitz operators are a class of operators that arise in the study of analytic functions (Hardy spaces)
• Pseudodifferential operators generalize differential operators and have applications in partial differential equations
• Operator theory finds applications in quantum mechanics, signal processing, and partial differential equations