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Trace class

In functional analysis, a trace class operator is a bounded linear operator T on a separable Hilbert space H for which the trace norm \|T\|_1 = \operatorname{tr}(|T|) = \sum_{n=1}^\infty \langle |T| e_n, e_n \rangle < \infty, where \{e_n\} is any orthonormal basis of H and |T| = (T^* T)^{1/2} is the absolute value of T.^[1] This condition ensures that the operator admits a well-defined, basis-independent trace \operatorname{tr}(T) = \sum_{n=1}^\infty \langle T e_n, e_n \rangle, generalizing the familiar trace from finite-dimensional linear algebra to infinite-dimensional settings.^[2] The space of all such operators, denoted B_1(H), forms a Banach space under the trace norm and is a two-sided *-ideal in the algebra B(H) of bounded operators on H.^[3] Trace class operators are inherently compact, as the finiteness of the trace norm implies that T can be approximated by finite-rank operators in the operator norm.^[1] They play a central role in operator theory, serving as the dual space of the compact operators K(H) under the duality pairing \langle T, A \rangle = \operatorname{tr}(T A) for T \in B_1(H) and A \in K(H), and as the predual of B(H) equipped with the operator norm.^[3] Products of trace class and bounded operators remain trace class, with the trace satisfying the cyclic property \operatorname{tr}(AB) = \operatorname{tr}(BA) whenever both products are defined.^[2] The concept originated in the mid-20th century, building on Alexander Grothendieck's 1955 work on nuclear operators and projective tensor products of Banach spaces, where he established conditions for traces on such operators.^[2] Victor Lidskii's 1959 theorem further connected the trace to the sum of eigenvalues, providing a spectral characterization essential for applications in quantum mechanics and spectral theory.^[2] Earlier contributions by John von Neumann and Robert Schatten in the 1940s and 1950s linked trace class operators to Hilbert-Schmidt operators and nuclearity, solidifying their equivalence on Hilbert spaces.^[2] These developments have made trace class operators indispensable in areas such as noncommutative geometry, where they facilitate index theory and determinant definitions for elliptic operators.^[3]

Fundamentals

Definition

In operator theory, trace-class operators are defined on separable Hilbert spaces and form an important ideal within the algebra of bounded linear operators. Let H be a separable Hilbert space, and let B(H) denote the C^*-algebra of bounded linear operators on H. For a bounded linear operator T \in B(H), the absolute value |T| is the unique positive square root of T^* T, i.e., |T| = (T^* T)^{1/2}. The trace functional \operatorname{Tr} on positive operators A \geq 0 in B(H) is given by

\operatorname{Tr}(A) = \sum_{k=1}^\infty \langle A e_k, e_k \rangle,

where \{e_k\}_{k=1}^\infty is any orthonormal basis of H; this sum converges absolutely and is independent of the basis chosen. An operator T \in B(H) is trace-class if \operatorname{Tr}(|T|) < \infty. The trace norm on trace-class operators is defined as \|T\|_1 = \operatorname{Tr}(|T|). This defines a norm on the space B_1(H) of all trace-class operators, satisfying positivity (\|T\|_1 \geq 0, with equality if and only if T = 0), homogeneity (\|c T\|_1 = |c| \|T\|_1 for scalars c), and the triangle inequality (\|T + S\|_1 \leq \|T\|_1 + \|S\|_1 for T, S \in B_1(H)). The space B_1(H), equipped with the trace norm, forms a Banach space and is a two-sided ideal in B(H). All trace-class operators are compact.

Equivalent formulations

Trace-class operators on a separable Hilbert space admit several equivalent characterizations, each providing insight into their structure and norm. These formulations extend the finite-dimensional notion of the trace, where the trace norm is the sum of singular values, to infinite dimensions by ensuring summability conditions on decompositions or spectra. One equivalent condition is that an operator T is nuclear, meaning it admits a decomposition T = \sum_{i=1}^\infty u_i \otimes v_i for sequences \{u_i\}, \{v_i\} in the Hilbert space such that \|T\|_1 = \inf \sum_{i=1}^\infty \|u_i\| \|v_i\| < \infty, where the infimum is over all such representations.^[4] A second characterization is that T factors as a product T = AB of two Hilbert-Schmidt operators A and B, with \|T\|_1 = \inf \|A\|_2 \|B\|_2 < \infty, the infimum taken over all such factorizations.^[5] Third, the singular values \{\mu_n(T)\}_{n=1}^\infty of T belong to the sequence space \ell^1, i.e., \sum_{n=1}^\infty \mu_n(T) < \infty, and this sum equals the trace norm \|T\|_1.^[3] Fourth, the absolute value |T| = \sqrt{T^* T} has finite trace, meaning \operatorname{tr}(|T|) = \sum_{n=1}^\infty \langle |T| e_n, e_n \rangle < \infty for any orthonormal basis \{e_n\}, and this trace equals \|T\|_1. Equivalently, |T|^{1/2} is Hilbert-Schmidt.^[3] Fifth, the adjoint T^* is also trace-class, with \|T^*\|_1 = \|T\|_1, since |T^*| = |T|.^[4] Sixth, for any bounded operators K and L, the composition K T L is trace-class with \|K T L\|_1 \leq \|K\| \|L\| \|T\|_1 < \infty. The equivalence between the nuclear and trace-norm definitions relies on the singular value decomposition (SVD). For a compact operator T, the SVD yields T = \sum_{n=1}^\infty \mu_n \langle \cdot, e_n \rangle f_n, where \{e_n\} and \{f_n\} are orthonormal bases of right and left singular vectors, respectively, and \mu_n are the singular values in decreasing order. This provides a nuclear decomposition with \sum \mu_n \|e_n\| \|f_n\| = \sum \mu_n < \infty, so every trace-class operator (with summable singular values) is nuclear, and the trace norm bounds the nuclear norm. Conversely, any nuclear decomposition satisfies \sum \|u_i\| \|v_i\| \geq \sum \mu_n by properties of the SVD and minimax principles for singular values, establishing equality of norms. The polar decomposition T = U |T|, with U partial isometry, further links the trace to the nuclear form via the spectral theorem applied to the positive operator |T|.^[4] These formulations generalize the matrix trace from finite dimensions, where the trace norm coincides with the sum of absolute eigenvalues for normal matrices or singular values otherwise, to infinite dimensions by replacing finite sums with absolutely convergent series, ensuring well-defined traces despite potential non-normality or lack of diagonalizability. In finite dimensions, all bounded operators are trace-class, but the infinite-dimensional conditions impose strict compactness and summability, distinguishing trace-class from broader classes like Hilbert-Schmidt operators.^[3]

Examples

Finite-rank operators

Finite-rank operators are bounded linear operators on a Hilbert space H whose range is finite-dimensional. Such operators can be expressed as T = \sum_{i=1}^n u_i \otimes v_i, where each u_i \otimes v_i is a rank-one operator defined by (u_i \otimes v_i)x = \langle x, v_i \rangle u_i for x \in H, and \{u_1, \dots, u_n\} forms a basis for the range of T.^[6] Every finite-rank operator is trace class because its singular value decomposition consists of only finitely many nonzero singular values \sigma_1(T), \dots, \sigma_r(T) (with r \leq n), and the trace norm is \|T\|_1 = \operatorname{Tr}(|T|) = \sum_{i=1}^r \sigma_i(T) < \infty.^[6] This finite sum ensures the operator belongs to the trace class ideal B_1(H).^[1] The trace of a finite-rank operator T = \sum_{i=1}^n u_i \otimes v_i can be computed explicitly as \operatorname{Tr}(T) = \sum_{i=1}^n \langle u_i, v_i \rangle. For the rank-one case T = u \otimes v, this reduces to \operatorname{Tr}(T) = \langle u, v \rangle, which follows from the basis-independent definition \operatorname{Tr}(T) = \sum_{k} \langle T e_k, e_k \rangle over any orthonormal basis \{e_k\}.^[7]^[3] Finite-rank operators are dense in the trace class B_1(H) under the trace norm \|\cdot\|_1; any trace-class operator can be approximated by truncating its singular value decomposition to finite rank, with the error bounded by the tail of the singular value sum.^[1] Every finite-rank operator is compact, as its range is finite-dimensional, and nuclear, since on Hilbert spaces the trace class coincides with the nuclear operators. The trace norm \|T\|_1 equals the sum of its singular values, providing a direct measure of its "size" in this context.^[1]^[6]

Self-adjoint operators via spectral theorem

The spectral theorem provides a fundamental decomposition for self-adjoint operators on a separable Hilbert space H. For a bounded self-adjoint operator T, there exists a unique projection-valued measure E, called the spectral resolution of T, supported on the spectrum \sigma(T) \subseteq \mathbb{R}, such that

T = \int_{\sigma(T)} \lambda \, dE(\lambda),

where the integral is understood in the weak operator topology.^[8] For compact self-adjoint operators, the spectrum consists of a countable set of real eigenvalues \{\lambda_n\} (counting multiplicities) accumulating only at zero, and the spectral measure is atomic: E is a sum of rank-one projections onto the corresponding eigenspaces. In this case, T is trace class if and only if \sum_n |\lambda_n| < \infty, and the trace satisfies \operatorname{Tr}(T) = \sum_n \lambda_n.^[9] A concrete illustration arises with diagonal operators on \ell^2(\mathbb{N}). Let T be the multiplication operator defined by (T x)_n = a_n x_n for x = (x_n) \in \ell^2(\mathbb{N}), where (a_n) is a real sequence in \ell^1(\mathbb{N}). Then T is self-adjoint and compact, with eigenvalues a_n (each of multiplicity one), so T is trace class with \operatorname{Tr}(T) = \sum_n a_n.^[9] For positive self-adjoint operators, the trace class condition simplifies further. If T \geq 0, then T is trace class if and only if \sum_k \langle T e_k, e_k \rangle < \infty for some (equivalently, any) orthonormal basis \{e_k\} of H, in which case \operatorname{Tr}(T) = \sum_k \langle T e_k, e_k \rangle. This equality between the trace and the sum of eigenvalues follows directly from the spectral theorem in the self-adjoint case and anticipates Lidskii's theorem, which establishes it for trace class operators more generally.

Integral operators via Mercer's theorem

Mercer's theorem provides a key construction of trace-class integral operators on Hilbert spaces of square-integrable functions over compact domains. Specifically, consider a continuous symmetric positive definite kernel K: [a, b] \times [a, b] \to \mathbb{R} on a compact interval [a, b]. The associated integral operator T on L^2([a, b]) is defined by

(Tf)(x) = \int_a^b K(x, y) f(y) \, dy

for f \in L^2([a, b]). Under these conditions, T is compact and self-adjoint, admitting an orthonormal basis of eigenfunctions \{\phi_n\} with corresponding positive eigenvalues \{\lambda_n\} satisfying the Mercer expansion

K(x, y) = \sum_{n=1}^\infty \lambda_n \phi_n(x) \phi_n(y),

where the series converges uniformly.^[10] The eigenvalues \lambda_n are absolutely summable, with \sum_{n=1}^\infty \lambda_n = \int_a^b K(x, x) \, dx < \infty, since K(x, x) is continuous on the compact set [a, b] and thus bounded. This finiteness of the eigenvalue sum implies that T is trace class. The trace of T is explicitly given by

\operatorname{Tr}(T) = \sum_{n=1}^\infty \lambda_n = \int_a^b K(x, x) \, dx.[](https://www.jstor.org/stable/2047610) Representative examples include Gaussian kernels, such as $K(x, y) = \exp\left( -\frac{(x - y)^2}{2\sigma^2} \right)$ for $\sigma > 0$, which are continuous, symmetric, and positive definite on any compact [interval](/page/Interval) $[a, b]$. The resulting operator $T$ is thus trace class, with trace $\operatorname{Tr}(T) = (b - a) \int_{-\infty}^\infty \exp\left( -\frac{u^2}{2\sigma^2} \right) \, du / \sqrt{2\pi} \sigma$, reflecting the [normalization](/page/Normalization) of the Gaussian [density](/page/Density). Similarly, polynomial kernels like $K(x, y) = 1 + x y$ on $[-1, 1]$ yield trace-class operators, as they satisfy the required conditions and produce explicitly computable Mercer expansions via orthogonal polynomials.[](https://arxiv.org/pdf/2106.08443)[](https://www.math.pku.edu.cn/teachers/yaoy/publications/MinNiyYao06.pdf) More broadly, while square-integrable kernels $K \in L^2([a, b] \times [a, b])$ guarantee that the [integral operator](/page/Integral_operator) is Hilbert-Schmidt (a larger class containing trace-class operators), achieving trace class requires stricter conditions, such as [continuity](/page/Continuity) and [positive definiteness](/page/Positive_definiteness) ensuring the diagonal integral bounds the eigenvalue sum. ## Properties ### Lidskii's theorem and trace properties The trace functional on the space of trace-class operators is linear, meaning that for any trace-class operators $T_1, T_2$ and scalars $\alpha, \beta$, $\operatorname{tr}(\alpha T_1 + \beta T_2) = \alpha \operatorname{tr}(T_1) + \beta \operatorname{tr}(T_2)$. It also satisfies the cyclic property: if $A$ is a [bounded operator](/page/Bounded_operator) and $B$ is trace-class, then $\operatorname{tr}(AB) = \operatorname{tr}(BA)$. Moreover, the trace is continuous with respect to the [trace](/page/Trace) [norm](/page/Norm), as $|\operatorname{tr}(T)| \leq \|T\|_1$ for any trace-class operator $T$. A central result concerning the [trace](/page/Trace) is Lidskii's theorem, which establishes that for any trace-class operator $T$ on a [Hilbert space](/page/Hilbert_space), the trace equals the sum of its eigenvalues (counted with algebraic multiplicity): \[ \operatorname{tr}(T) = \sum_n \lambda_n(T),

where the sum converges absolutely. The proof proceeds by decomposing T into its self-adjoint and skew-adjoint parts, applying the holomorphic functional calculus to express the eigenvalues via the Riesz integral representation, and verifying the equality through contour integration over suitable paths enclosing the spectrum. Additional properties include the fact that the trace is real on self-adjoint operators and satisfies \operatorname{tr}(T^*) = \overline{\operatorname{tr}(T)}. For self-adjoint operators, the trace is real-valued. The set of trace-class operators with zero trace forms a codimension-1 subspace of the trace-class ideal. Lidskii's theorem, originally proved in 1959, resolved longstanding questions about the relationship between the trace functional and eigenvalue sums for non-self-adjoint trace-class operators.

Relationships with other operator classes

Trace-class operators occupy a distinguished position within the hierarchy of operator ideals on a Hilbert space H. They form a proper subclass of the Hilbert–Schmidt operators, which are themselves properly contained in the compact operators, and these in turn are properly contained in the bounded operators: S_1(H) \subsetneq S_2(H) \subsetneq K(H) \subsetneq B(H).^[11] For any operator T belonging to these classes, the Schatten norms satisfy \|T\|_1 \geq \|T\|_2 \geq \|T\|, where \|T\| denotes the operator norm.^[11] The trace class S_1(H) is the case p=1 of the more general Schatten p-classes S_p(H) for $1 \leq p \leq \infty, defined as the ideal of compact operators T whose singular values \{\mu_n(T)\}_{n=1}^\infty (arranged in decreasing order) belong to the sequence space \ell^p, equipped with the norm

\|T\|_p = \left( \sum_{n=1}^\infty \mu_n(T)^p \right)^{1/p}

for $1 \leq p < \infty, and \|T\|_\infty = \sup_n \mu_n(T) = \|T\| for p = \infty, where S_\infty(H) = K(H).^[11] These classes satisfy the inclusions S_p(H) \subset S_q(H) for $1 \leq p < q \leq \infty, with continuous embeddings \|T\|_q \leq \|T\|_p.^[11] On Hilbert spaces, the Schatten classes analogize the \ell^p spaces via the singular value sequences, providing a non-commutative counterpart to classical sequence space theory.^[11] In particular, trace-class operators coincide with the nuclear operators on H, which are those admitting a factorization T = \sum_k u_k \otimes v_k with \sum_k \|u_k\| \|v_k\| < \infty.^[12] The trace class enjoys the two-sided ideal property within the bounded operators: if A \in B(H) and T \in S_1(H), then AT, TA \in S_1(H), with the submultiplicative estimate \|AT\|_1 \leq \|A\| \|T\|_1.^[1] This extends to the full Schatten classes, which are two-sided ideals in B(H).^[1] Regarding density, the finite-rank operators are dense in the trace class with respect to the trace norm \|\cdot\|_1.^[5] Similarly, the Hilbert–Schmidt operators are dense in the compact operators with respect to the operator norm, as they contain the finite-rank operators, which form a dense subspace of K(H).^[3]

Dual space characterizations

The trace class operators on a Hilbert space H, denoted B_1(H), form a Banach space under the trace norm and stand in duality with the space of compact operators K(H). Specifically, the continuous dual space K(H)^* is isometrically isomorphic to B_1(H) via the pairing \langle T, A \rangle = \operatorname{tr}(T A^*) for T \in K(H) and A \in B_1(H).^[13] This pairing is well-defined because the product T A^* is trace class whenever T is compact and A is trace class, and the trace norm on B_1(H) coincides with the dual norm induced by the operator norm on K(H). The isomorphism arises from the nuclearity of Hilbert spaces, ensuring that every bounded linear functional on K(H) extends uniquely to a trace class operator representation.^[13] A proof outline relies on the Riesz representation theorem adapted to operator spaces: any continuous functional \phi \in K(H)^* can be expressed as \phi(T) = \operatorname{tr}(T B) for some B \in B_1(H), where B is constructed via singular value decompositions or approximations by finite-rank operators, establishing the isometry. This result originates from Grothendieck's characterization of nuclear operators in Hilbert spaces, where trace class operators precisely capture the dual structure of compacts.^[13] The equivalence of norms follows from the inequality |\operatorname{tr}(T A^*)| \leq \|T\| \|A\|_1, with equality achieved in the dense subspace of finite-rank operators. Furthermore, B_1(H) serves as the predual of the space of bounded operators B(H), meaning B(H) is isometrically isomorphic to the dual of B_1(H) when B(H) is equipped with the weak* topology. In this duality, every weak* continuous linear functional on B(H) arises from a trace class operator via the same trace pairing, and B_1(H) embeds densely into B(H)^* under the norm topology, though the full dual B(H)^* is larger. This predual relationship highlights the role of trace class operators in completing the dual pair structure for B(H), particularly in the context of von Neumann algebra theory. The implications of these dualities extend to the C*-algebra structure: K(H) is a closed ideal in B(H), and identifying its dual as B_1(H) provides a Banach space completion that aligns with the multiplier algebra framework, facilitating representations of states and traces in operator theory.^[13]

Applications

In quantum mechanics

In quantum mechanics, trace-class operators play a central role in the description of quantum states, particularly through the formalism of density matrices. A density matrix \rho is defined as a self-adjoint, positive semi-definite trace-class operator on a Hilbert space \mathcal{H} satisfying \operatorname{Tr}(\rho) = 1.^[14] These operators represent mixed quantum states, which arise when the system is in an ensemble of pure states or interacts with an environment, generalizing the concept of pure states |\psi\rangle\langle\psi| to statistical mixtures.^[14] Unlike pure states, density matrices capture incomplete knowledge or decoherence effects inherent in real quantum systems.^[15] The expectation value of a bounded observable A, represented by a self-adjoint operator on \mathcal{H}, is computed as \langle A \rangle = \operatorname{Tr}(\rho A).^[14] This trace formula provides a probabilistic interpretation, where the diagonal elements of \rho in the eigenbasis of A yield the probabilities of measurement outcomes.^[15] A key measure of the mixedness of a state is the purity \operatorname{Tr}(\rho^2), which satisfies $0 < \operatorname{Tr}(\rho^2) \leq 1, with equality to 1 indicating a pure state and values approaching $1/\dim(\mathcal{H}) for maximally mixed states.^[16] For composite quantum systems on \mathcal{H}_A \otimes \mathcal{H}_B, the partial trace \operatorname{Tr}_B(\rho_{AB}) over subsystem B produces the reduced density operator \rho_A for subsystem A, which remains a trace-class operator with \operatorname{Tr}(\rho_A) = 1.^[15] This operation is essential for quantifying entanglement, as the eigenvalues of \rho_A reveal correlations between A and B; for separable states, \rho_A decomposes into a convex combination of product states, while entangled states exhibit non-classical features in their reduced descriptions.^[15] Post-2010 developments have extended trace-class operators to open quantum systems and quantum information theory, notably through the trace-norm distance \|\rho - \sigma\|_1 = \operatorname{Tr}(|\rho - \sigma|), which quantifies the maximum distinguishability of two states \rho and \sigma under optimal measurements.^[17] In open systems, trace-preserving completely positive maps model dissipative dynamics while preserving the trace-class structure of density operators.^[18] These tools underpin advances in quantum error correction and state tomography, where the trace norm bounds error propagation in noisy channels.^[17]

In numerical analysis

In numerical analysis, trace-class operators play a crucial role in computational tasks involving large-scale linear algebra, particularly for approximating traces and trace norms when direct computation is infeasible due to the operator's dimension or implicit representation. These operators, characterized by a finite sum of singular values, admit efficient randomized approximation methods that exploit spectral decay for scalability. Such techniques are essential for applications in scientific computing, where operators arise from discretizations of integral equations or kernel matrices. A prominent method for trace estimation is Hutchinson's estimator, which provides an unbiased approximation of the trace \operatorname{Tr}(A) for a symmetric matrix A (or more generally, a self-adjoint operator) via

\operatorname{Tr}(A) \approx \frac{1}{m} \sum_{i=1}^m \langle v_i, A v_i \rangle,

where \{v_i\}_{i=1}^m are independent random vectors satisfying \mathbb{E}[v v^T] = I, such as standard Gaussian or Rademacher vectors.^[19] This estimator requires only matrix-vector products, making it suitable for implicit trace-class operators accessed via linear actions. For positive semidefinite A, the variance is bounded by \frac{2}{m} \operatorname{Tr}(A^2) \leq \frac{2}{m} \|A\|_F^2, with modern analyses providing sharper concentration guarantees under spectral assumptions typical of trace-class operators, achieving relative error \epsilon with O(\|A\|_F^2 / (\epsilon^2 \operatorname{Tr}(A)^2)) probes in the worst case.^[20] To approximate the trace norm \|T\|_1 = \sum_i \sigma_i(T) of a compact operator T, methods focus on capturing the dominant singular values, leveraging the rapid decay inherent to trace-class spectra. The Lanczos bidiagonalization algorithm iteratively constructs an orthonormal basis for the range of T and T^*, yielding a small bidiagonal matrix whose singular values approximate the largest of T, enabling partial sums as proxies for the full trace norm with controlled error for decaying spectra.^[21] Complementarily, randomized SVD techniques project T onto a low-dimensional random subspace, computing a truncated decomposition that approximates \|T\|_1 efficiently for large sparse operators, with expected error bounded by factors involving the tail singular values and oversampling parameter p \geq 2.^[22] These approaches scale linearly in the rank while requiring O(k \log k) iterations for k-term approximations, outperforming dense SVD for high-dimensional problems. In kernel methods for machine learning, trace-class properties of covariance operators from Mercer kernels underpin efficient computation in Gaussian processes, where the effective dimension—defined as \operatorname{Tr}((K + \lambda I)^{-1} K) for kernel matrix K and regularization \lambda > 0—remains finite, quantifying model complexity and enabling generalization bounds like O(\sqrt{d_{\text{eff}} / n}) excess risk for n samples. This finiteness arises because trace-class kernels induce operators with absolutely summable eigenvalues, controlling the degrees of freedom in regression or classification tasks without overfitting in high dimensions. Mercer's theorem ensures such kernels correspond to positive definite integral operators that are trace-class under suitable decay conditions. Recent advances in the 2020s have introduced quantum-inspired algorithms for trace estimation on classical hardware, adapting dequantization techniques like randomized tensor sketches to estimate multivariate traces in high-dimensional settings, achieving near-quadratic speedups over classical baselines for structured operators while remaining fully implementable on GPUs.^[23] These methods address scalability challenges in simulating quantum-like linear algebra tasks, such as log-determinant approximations in optimization, by exploiting low-rank structure analogous to trace-class decay.

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