---
id: wiki-2026-0508-operator-theory
title: Operator Theory
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Operator-Theory, 작용소-이론, Functional-Analysis-Operators]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [math, functional-analysis, linear-algebra, hilbert-space, spectral-theory]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: scipy-linalg
---

# Operator Theory

## 매 한 줄
> **"매 행렬을 무한차원으로 확장한 게 operator"**. Hilbert/Banach 공간의 linear map을 다루며, 매 spectral theorem·compact operators·C*-algebras를 통해 quantum mechanics·PDE·signal processing을 통합한다. 2026 ML에서는 매 Koopman operator로 dynamical system 학습에 부활.

## 매 핵심

### 매 분류
- **Bounded vs Unbounded**: 매 norm 제한 여부.
- **Compact**: 매 unit ball을 relatively compact set으로 보내는 operator.
- **Self-adjoint**: T = T*; 매 eigenvalue 실수.
- **Unitary**: T*T = I; 매 inner product 보존.
- **Normal**: TT* = T*T; 매 spectral theorem 적용 가능.

### 매 spectral 분해
- Finite-dim: eigenvalue decomposition.
- Compact self-adjoint: countable eigenvalues + ON eigenvectors (Hilbert-Schmidt).
- Bounded self-adjoint: spectral measure dE(λ); T = ∫λ dE(λ).

### 매 응용
1. Quantum mechanics — observable = self-adjoint operator.
2. PDE — Laplacian, Schrödinger evolution.
3. Koopman/Perron-Frobenius — dynamical system linearization.
4. RKHS / kernel methods — integral operator.

## 💻 패턴

### Bounded operator (Python class)
```python
import numpy as np
class IntegralOperator:
    def __init__(self, kernel, grid):
        self.K = kernel(grid[:, None], grid[None, :]) * (grid[1]-grid[0])
    def __call__(self, f):
        return self.K @ f
```

### Self-adjointness check
```python
def is_self_adjoint(A, tol=1e-10):
    return np.allclose(A, A.conj().T, atol=tol)
```

### Spectral decomposition
```python
A = np.array([[2., 1.], [1., 3.]])
w, V = np.linalg.eigh(A)  # self-adjoint → real eigenvalues
# A = V diag(w) V^T
```

### Functional calculus (matrix exp)
```python
from scipy.linalg import expm
U = expm(-1j * H * t)  # quantum time evolution
```

### Koopman operator (data-driven)
```python
def dmd(X, Xp, r=10):
    U, S, Vt = np.linalg.svd(X, full_matrices=False)
    Ur, Sr, Vr = U[:, :r], S[:r], Vt[:r].conj().T
    A_tilde = Ur.conj().T @ Xp @ Vr / Sr
    eigs, W = np.linalg.eig(A_tilde)
    modes = Xp @ Vr / Sr @ W
    return eigs, modes
```

### RKHS kernel as integral operator
```python
def rkhs_eval(alpha, X, x_new, kernel):
    return sum(a * kernel(xi, x_new) for a, xi in zip(alpha, X))
```

## 매 결정 기준
| 문제 | Operator class |
|---|---|
| Quantum observable | Self-adjoint, possibly unbounded |
| Markov chain | Stochastic (positivity-preserving) |
| Signal filter | Bounded, possibly unitary |
| Dynamics learning | Koopman (compact approx) |

**기본값**: Self-adjoint compact (가장 잘 분석됨).

## 🔗 Graph
- 부모: [[Linear-Algebra-Foundations]]
- Adjacent: [[Hilbert-Space]] · [[Eigenvalues-and-Eigenvectors]]

## 🤖 LLM 활용
**언제**: Infinite-dim linear systems, PDE/quantum modeling, operator-learning (FNO, DeepONet).
**언제 X**: Pure finite-dim — 매 그냥 matrix theory로 충분.

## ❌ 안티패턴
- **Unbounded operator를 bounded로 가정**: 매 domain 무시.
- **Non-normal에 spectral theorem**: 매 잘못 적용.
- **Truncation 후 boundary effect 무시**.

## 🧪 검증 / 중복
- Verified (Reed & Simon "Methods of Modern Mathematical Physics" Vol. 1).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — Operator classes + Koopman pattern |
