QuantumInfo.ForMathlib.HayataGroup.TraceInequality.HilbertSchmidtOperatorSpace

Given a finite-dimensional complex Hilbert space $\mathcal{H}$, the indexing set $\text{HSIndex}(\mathcal{H})$ is defined as the set of natural numbers $\{0, 1, \dots, n-1\}$, where $n = \dim_{\mathbb{C}} \mathcal{H}$ is the complex dimension of $\mathcal{H}$. This set serves as the canonical indexing set for Hilbert-Schmidt coordinates.

Given a finite-dimensional complex Hilbert space $\mathcal{H}$ with complex dimension $n$, this definition provides the standard orthonormal basis for $\mathcal{H}$. The basis is indexed by the set $\text{HSIndex}(\mathcal{H}) = \{0, 1, \dots, n-1\}$.

Given a finite-dimensional complex Hilbert space $\mathcal{H}$, the Hilbert-Schmidt coordinate space is defined as the Euclidean space $\mathbb{C}^{\text{HSIndex}(\mathcal{H}) \times \text{HSIndex}(\mathcal{H})}$. Here, $\text{HSIndex}(\mathcal{H})$ is the canonical indexing set $\{0, 1, \dots, n-1\}$ where $n = \dim_{\mathbb{C}} \mathcal{H}$. This space consists of complex-valued functions on the product of the index set with itself, effectively representing Hilbert-Schmidt operators on $\mathcal{H}$ as $n \times n$ matrices equipped with the $L^2$ (Frobenius) norm.

For a finite-dimensional complex Hilbert space $\mathcal{H}$ with dimension $n = \dim_{\mathbb{C}} \mathcal{H}$, let $I = \{0, 1, \dots, n-1\}$ be the canonical indexing set. The Hilbert-Schmidt coordinate function space is the space of functions from the Cartesian product $I \times I$ to the complex numbers $\mathbb{C}$. This space represents the underlying coordinate system for Hilbert-Schmidt operators acting on $\mathcal{H}$.

Given a complex inner product space $\mathcal{H}$, the space $HSOp(\mathcal{H})$ is defined as the space of bounded linear operators $\mathcal{B}(\mathcal{H})$ on $\mathcal{H}$. This definition is intended to be equipped with the Hilbert-Schmidt structure (the Hilbert-Schmidt inner product and norm).

For a complex inner product space $\mathcal{H}$, the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ forms an additive commutative group.

For a complex inner product space $\mathcal{H}$, the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ (defined as the space of bounded linear operators $\mathcal{B}(\mathcal{H})$) is a module over the complex numbers $\mathbb{C}$, making it a complex vector space.

For a complex inner product space $\mathcal{H}$, the space $HSOp(\mathcal{H})$ (representing the bounded linear operators $\mathcal{B}(\mathcal{H})$) is equipped with a star operation. This operation maps an operator $A \in HSOp(\mathcal{H})$ to its adjoint operator $A^*$.

For a complex inner product space $\mathcal{H}$, the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ (defined as the space of bounded linear operators $\mathcal{B}(\mathcal{H})$) is equipped with a multiplication operation. This multiplication is defined by the composition of operators, where for any two operators $A, B \in HSOp(\mathcal{H})$, their product $A \cdot B$ is the composition $A \circ B$.

For a complex inner product space $\mathcal{H}$, this instance defines the multiplicative identity element $1$ in the Hilbert-Schmidt operator space $HSOp(\mathcal{H})$. This element corresponds to the identity operator $I$ on $\mathcal{H}$.

Given a complex inner product space $\mathcal{H}$, the space of Hilbert-Schmidt operators $\text{HSOp}(\mathcal{H})$ is inhabited, meaning it contains at least one element (specifically, the zero operator).

For a complex inner product space $\mathcal{H}$, the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ is equipped with a zero element, which corresponds to the zero operator that maps every vector in $\mathcal{H}$ to the zero vector.

Let $\mathcal{H}$ be a finite-dimensional complex inner product space. The space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ is a finite-dimensional vector space over the complex numbers $\mathbb{C}$.

For a finite-dimensional complex Hilbert space $\mathcal{H}$ with dimension $n = \dim_{\mathbb{C}} \mathcal{H}$, let $\text{HSIndex}(\mathcal{H}) = \{0, 1, \dots, n-1\}$ be its canonical indexing set. This definition provides a $\mathbb{C}$-linear equivalence between the Hilbert-Schmidt coordinate space $\text{HSCoords}(\mathcal{H})$ (the space of complex-valued functions on $\text{HSIndex}(\mathcal{H}) \times \text{HSIndex}(\mathcal{H})$ equipped with the $L^2$ norm) and the Hilbert-Schmidt coordinate function space $\text{HSCoordFun}(\mathcal{H})$ (the underlying space of complex-valued functions on the same index set).

For a finite-dimensional complex Hilbert space $\mathcal{H}$ with dimension $n = \dim_{\mathbb{C}} \mathcal{H}$, let $I = \{0, 1, \dots, n-1\}$ be the canonical index set. This definition provides a $\mathbb{C}$-linear equivalence between the space of matrices $\text{Matrix}(I, I, \mathbb{C})$ and the coordinate function space $I \times I \to \mathbb{C}$. Specifically, it identifies a matrix $M$ with the function $f(i, j) = M_{i,j}$.

For a finite-dimensional complex Hilbert space $\mathcal{H}$ with dimension $n = \dim_{\mathbb{C}} \mathcal{H}$, let $I = \{0, 1, \dots, n-1\}$ be the canonical index set. This definition provides a $\mathbb{C}$-linear equivalence between the space of complex matrices $\text{Matrix}(I, I, \mathbb{C})$ and the Hilbert-Schmidt coordinate space $\text{HSCoords}(\mathcal{H})$. It is obtained by mapping a matrix to its corresponding function representation and then into the Hilbert-Schmidt coordinate space.

For a complex Hilbert space $\mathcal{H}$, this definition provides a $\mathbb{C}$-linear equivalence between the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ (defined as the space of bounded linear operators $\mathcal{B}(\mathcal{H})$) and the space of all $\mathbb{C}$-linear endomorphisms on $\mathcal{H}$ (denoted $\mathcal{H} \to_{\mathbb{C}} \mathcal{H}$ or $\text{End}_{\mathbb{C}}(\mathcal{H})$). This equivalence identifies a continuous operator with its underlying linear map, effectively "forgetting" the continuity requirement.

For a finite-dimensional complex Hilbert space $\mathcal{H}$, this definition provides a $\mathbb{C}$-linear equivalence between the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ and the Hilbert-Schmidt coordinate space $\text{HSCoords}(\mathcal{H}) \cong \mathbb{C}^{I \times I}$, where $I = \{0, 1, \dots, n-1\}$ is the canonical index set and $n = \dim_{\mathbb{C}} \mathcal{H}$. This equivalence maps an operator $T \in HSOp(\mathcal{H})$ to its matrix representation relative to the standard orthonormal basis $\{e_i\}_{i \in I}$ of $\mathcal{H}$. Specifically, for an operator $T$, its coordinate at index $(i, j)$ corresponds to the matrix entry $M_{ij}$ of the linear map underlying $T$ with respect to the basis.

Let $\mathcal{H}$ be a finite-dimensional complex Hilbert space and $I = \{0, 1, \dots, n-1\}$ be its canonical index set, where $n = \dim_{\mathbb{C}} \mathcal{H}$. Let $\mathcal{B}$ be the standard orthonormal basis for $\mathcal{H}$. For any Hilbert-Schmidt operator $T \in HSOp(\mathcal{H})$ and any indices $i, j \in I$, the $(i, j)$-th component of the coordinate representation of $T$ in the space $\text{HSCoords}(\mathcal{H}) \cong \mathbb{C}^{I \times I}$ is equal to the $(i, j)$-th entry of the matrix representing the linear map $T$ with respect to the basis $\mathcal{B}$.

For a finite-dimensional complex Hilbert space $\mathcal{H}$, the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ is a normed additive commutative group. The norm on $HSOp(\mathcal{H})$ is induced from the Euclidean (Frobenius) norm on the coordinate space $\text{HSCoords}(\mathcal{H}) \cong \mathbb{C}^{n \times n}$ via the canonical linear equivalence $\text{toHSCoordsLinearEquiv}$.

For a finite-dimensional complex Hilbert space $\mathcal{H}$, the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ is a normed space over the complex numbers $\mathbb{C}$. This normed space structure is induced from the norm on the coordinate space $\text{HSCoords}(\mathcal{H}) \cong \mathbb{C}^{n \times n}$ (where $n = \dim_{\mathbb{C}} \mathcal{H}$) via the canonical $\mathbb{C}$-linear equivalence $\text{toHSCoordsLinearEquiv}$.

Given a finite-dimensional complex Hilbert space $\mathcal{H}$, this definition equips the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ with an inner product. For any two operators $T, S \in HSOp(\mathcal{H})$, their inner product $\langle T, S \rangle$ is defined as the standard complex inner product of their matrix representations in the coordinate space $\text{HSCoords}(\mathcal{H}) \cong \mathbb{C}^{n \times n}$ (where $n = \dim_{\mathbb{C}} \mathcal{H}$). Effectively, if $M$ and $N$ are the matrices associated with $T$ and $S$ with respect to the canonical orthonormal basis, the inner product is the Frobenius inner product $\langle M, N \rangle_{\text{Frob}} = \sum_{i,j} \overline{M_{ij}} N_{ij}$.

Given a finite-dimensional complex Hilbert space $\mathcal{H}$, the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ is an inner product space over the complex numbers $\mathbb{C}$. This structure is induced by the Hilbert-Schmidt inner product (equivalent to the Frobenius inner product on the matrix representations) and is compatible with the normed space structure on $HSOp(\mathcal{H})$.

For a finite-dimensional complex Hilbert space $\mathcal{H}$, the map from the space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ to the coordinate space $HSCoords(\mathcal{H}) \cong \mathbb{C}^{n \times n}$ (where $n = \dim_{\mathbb{C}} \mathcal{H}$) is a $\mathbb{C}$-linear isometry equivalence. This means the map is a bijective linear transformation that preserves the Hilbert-Schmidt inner product, such that for any $T, S \in HSOp(\mathcal{H})$, $\langle T, S \rangle = \langle \text{coords}(T), \text{coords}(S) \rangle$, where the right-hand side is the standard Frobenius inner product on matrices.

Let $\mathcal{H}$ be a complex inner product space. The space of Hilbert-Schmidt operators $HSOp(\mathcal{H})$ is a complete space (i.e., every Cauchy sequence in $HSOp(\mathcal{H})$ converges to a limit within the space).

Let $\mathcal{H}$ be a complex inner product space. In the Hilbert-Schmidt operator space $HSOp(\mathcal{H})$, the scalar multiplication map $\mathbb{C} \times HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ is continuous with respect to the Hilbert-Schmidt norm.

For a complex inner product space $\mathcal{H}$, this function reinterprets a bounded linear operator $T \in \mathcal{B}(\mathcal{H})$ as an element of the Hilbert-Schmidt operator space $\text{HSOp}(\mathcal{H})$. This allows the operator to be treated in the context of the Hilbert-Schmidt inner product and norm.

For a complex inner product space $\mathcal{H}$, this function maps an element $T$ from the Hilbert-Schmidt operator space $HSOp(\mathcal{H})$ to its corresponding bounded linear operator in $\mathcal{B}(\mathcal{H})$. It effectively forgets the Hilbert-Schmidt inner product and norm structure to recover the underlying operator.

For a complex inner product space $\mathcal{H}$ and a bounded linear operator $A \in \mathcal{B}(\mathcal{H})$, this definition defines a continuous linear map from the Hilbert-Schmidt operator space $HSOp(\mathcal{H})$ to itself. The map takes a Hilbert-Schmidt operator $T$ and returns the operator $A T$ (the composition of $A$ and $T$), reinterpreted as an element of $HSOp(\mathcal{H})$.

For a bounded linear operator $B \in \mathcal{B}(\mathcal{H})$ on a complex inner product space $\mathcal{H}$, this definition represents the continuous linear map from the Hilbert-Schmidt operator space $HSOp(\mathcal{H})$ to itself, defined by right multiplication: $T \mapsto T B$ for any $T \in HSOp(\mathcal{H})$.

Let $\mathcal{H}$ be a complex inner product space. For any bounded linear operator $A \in \mathcal{B}(\mathcal{H})$ and any Hilbert-Schmidt operator $T \in HSOp(\mathcal{H})$, the bounded linear operator corresponding to the left multiplication of $T$ by $A$ (denoted $L_A(T)$) is equal to the operator composition $A T$.

Let $\mathcal{H}$ be a complex inner product space. For any bounded linear operator $B \in \mathcal{B}(\mathcal{H})$ and any Hilbert-Schmidt operator $T \in HSOp(\mathcal{H})$, the bounded linear operator corresponding to the application of the right multiplication map by $B$ to $T$ is equal to the composition of the operator corresponding to $T$ and $B$. That is, $\operatorname{toOp}(\operatorname{rightMulHS} B \, T) = (\operatorname{toOp} T) \circ B$.

Let $\mathcal{H}$ be a complex inner product space and let $\mathcal{B}(\mathcal{H})$ denote the space of bounded linear operators on $\mathcal{H}$. For any operator $A \in \mathcal{B}(\mathcal{H})$, let $\mathcal{L}_A: HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ be the continuous linear map that sends a Hilbert-Schmidt operator $T$ to $AT$. For any $A, B \in \mathcal{B}(\mathcal{H})$, the left multiplication operator corresponding to the product $AB$ is equal to the composition of the left multiplication operators for $A$ and $B$: $$\mathcal{L}_{AB} = \mathcal{L}_A \circ \mathcal{L}_B$$

Let $\mathcal{H}$ be a complex inner product space and $HSOp(\mathcal{H})$ be the space of Hilbert-Schmidt operators on $\mathcal{H}$. For any bounded linear operators $A, B \in \mathcal{B}(\mathcal{H})$, let $R_C: HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ denote the continuous linear map defined by right multiplication $R_C(T) = T C$. Then, the right multiplication operator by the product $AB$ satisfies $R_{AB} = R_B \circ R_A$, where $\circ$ denotes the composition of operators.

For a complex inner product space $\mathcal{H}$, let $1$ denote the identity operator on $\mathcal{H}$ and let $HSOp(\mathcal{H})$ denote the space of Hilbert-Schmidt operators on $\mathcal{H}$. The operator $\text{leftMulHS}(1)$, which represents left multiplication by the identity on $HSOp(\mathcal{H})$, is equal to the identity operator on $HSOp(\mathcal{H})$.

Let $\mathcal{H}$ be a complex inner product space. The right multiplication operator by the identity operator $1 \in \mathcal{B}(\mathcal{H})$ on the Hilbert-Schmidt operator space $HSOp(\mathcal{H})$, denoted as $\text{rightMulHS}(1)$, is equal to the identity operator on $HSOp(\mathcal{H})$.

For any bounded linear operators $A, B \in \mathcal{B}(\mathcal{H})$ on a complex inner product space $\mathcal{H}$, the left multiplication operator $L_A : HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ defined by $L_A(T) = AT$ and the right multiplication operator $R_B : HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ defined by $R_B(T) = TB$ commute. That is, $L_A \circ R_B = R_B \circ L_A$, or equivalently $A(TB) = (AT)B$ for all $T \in HSOp(\mathcal{H})$.

Let $\mathcal{H}$ be a complex inner product space. For any bounded linear operators $X, Y \in \mathcal{B}(\mathcal{H})$, the Hilbert-Schmidt inner product of $X$ and $Y$ is equal to the trace of the operator $X^* Y$, where $X^*$ denotes the adjoint of $X$. That is, $$\langle X, Y \rangle_{HS} = \operatorname{tr}(X^* Y)$$

Let $\mathcal{H}$ be a complex inner product space. For any bounded linear operators $X, Y \in \mathcal{B}(\mathcal{H})$, the real part of their Hilbert-Schmidt inner product $\langle X, Y \rangle_{\text{HS}}$ is equal to the real part of the trace of the operator $X^* Y$, where $X^*$ denotes the adjoint of $X$. That is, $$\text{Re}(\langle X, Y \rangle_{\text{HS}}) = \text{Re}(\text{Tr}(X^* Y)).$$

Let $\mathcal{H}$ be a complex inner product space. For any bounded linear operator $A \in \mathcal{B}(\mathcal{H})$, let $L_A: HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ be the left multiplication operator defined by $L_A(T) = AT$ for all $T \in HSOp(\mathcal{H})$. Then the adjoint of $L_A$ with respect to the Hilbert-Schmidt inner product is the left multiplication operator by the adjoint of $A$. That is, $$(L_A)^* = L_{A^*}$$ where $A^*$ is the adjoint of $A$ on $\mathcal{H}$ and $(L_A)^*$ is the adjoint of $L_A$ on the Hilbert-Schmidt operator space $HSOp(\mathcal{H})$.

Let $\mathcal{H}$ be a complex inner product space. Let $L(\mathcal{H})$ denote the space of bounded linear operators on $\mathcal{H}$, and let $HSOp(\mathcal{H})$ be the space of Hilbert-Schmidt operators on $\mathcal{H}$. For any operator $A \in L(\mathcal{H})$, let $\text{leftMulHS}(A)$ denote the continuous linear map on $HSOp(\mathcal{H})$ defined by $T \mapsto AT$. For any real number $r \in \mathbb{R}$, we have $$\text{leftMulHS}(r \cdot I_{\mathcal{H}}) = r \cdot I_{HSOp(\mathcal{H})}$$ where $I_{\mathcal{H}}$ is the identity operator on $\mathcal{H}$ and $I_{HSOp(\mathcal{H})}$ is the identity operator on the space $HSOp(\mathcal{H})$.

Let $\mathcal{H}$ be a complex inner product space and $HSOp(\mathcal{H})$ denote the space of Hilbert-Schmidt operators on $\mathcal{H}$. For any real number $r \in \mathbb{R}$, the right multiplication operator $R_{r I}: HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$, defined by $T \mapsto T(r I)$ where $I$ is the identity operator on $\mathcal{H}$, is equal to the scalar multiplication operator $r I_{HSOp(\mathcal{H})}$, where $I_{HSOp(\mathcal{H})}$ is the identity operator on $HSOp(\mathcal{H})$.

Let $\mathcal{H}$ be a complex inner product space. For any bounded linear operator $A \in \mathcal{B}(\mathcal{H})$ such that $A \ge 0$, the left multiplication operator $L_A: HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ (defined on the space of Hilbert-Schmidt operators by $L_A(T) = AT$) is also a non-negative operator ($L_A \ge 0$) with respect to the Hilbert-Schmidt inner product.

Let $\mathcal{H}$ be a complex inner product space. For any bounded linear operators $A, B \in \mathcal{B}(\mathcal{H})$, if $A \le B$ (in the sense of the Loewner order), then the left multiplication operators $L_A$ and $L_B$ acting on the Hilbert-Schmidt operator space $HSOp(\mathcal{H})$ (where $L_A(T) = AT$) satisfy $L_A \le L_B$.

Let $\mathcal{H}$ be a complex inner product space and $\mathcal{B}(\mathcal{H})$ be the space of bounded linear operators on $\mathcal{H}$. Let $HSOp(\mathcal{H})$ be the Hilbert space of Hilbert-Schmidt operators on $\mathcal{H}$. For any operator $A \in \mathcal{B}(\mathcal{H})$, let $L_A: HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ be the left multiplication operator defined by $L_A(T) = AT$. If $A$ is a strictly positive operator (i.e., $A$ is self-adjoint and its spectrum $\sigma(A)$ is contained in $(0, \infty)$), then $L_A$ is a strictly positive operator on $HSOp(\mathcal{H})$.

Let $\mathcal{H}$ be a complex inner product space and $HSOp(\mathcal{H})$ be the space of Hilbert-Schmidt operators on $\mathcal{H}$. For any bounded linear operator $A \in \mathcal{B}(\mathcal{H})$, let $R_A: HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ be the continuous linear map defined by right multiplication, $R_A(T) = TA$. Then the adjoint of this map with respect to the Hilbert-Schmidt inner product is the right multiplication map by the adjoint operator $A^*$, namely $(R_A)^* = R_{A^*}$.

Let $\mathcal{H}$ be a complex inner product space. This definition provides the real $\star$-algebra homomorphism from the algebra of bounded linear operators $\mathcal{B}(\mathcal{H})$ to the algebra of bounded linear operators on the Hilbert-Schmidt operator space, denoted $\mathcal{B}(HSOp(\mathcal{H}))$. For any operator $A \in \mathcal{B}(\mathcal{H})$, the homomorphism maps it to the continuous linear map $L_A: HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ defined by left multiplication, $L_A(T) = AT$ for any Hilbert-Schmidt operator $T \in HSOp(\mathcal{H})$.

Let $\mathcal{H}$ be a complex inner product space. Let $\mathcal{B}(\mathcal{H})$ denote the algebra of bounded linear operators on $\mathcal{H}$, and let $\mathcal{HS}(\mathcal{H})$ be the space of Hilbert-Schmidt operators on $\mathcal{H}$ (equipped with the Hilbert-Schmidt inner product). This definition provides the real $\star$-algebra homomorphism from the opposite algebra $\mathcal{B}(\mathcal{H})^{\text{op}}$ to the space of bounded linear operators acting on the Hilbert-Schmidt space, $\mathcal{B}(\mathcal{HS}(\mathcal{H}))$. Given an element $A \in \mathcal{B}(\mathcal{H})^{\text{op}}$, the homomorphism maps it to the continuous linear map $R_{A^{\text{unop}}}: \mathcal{HS}(\mathcal{H}) \to \mathcal{HS}(\mathcal{H})$ defined by right multiplication, $R_{A^{\text{unop}}}(T) = T A^{\text{unop}}$ for any $T \in \mathcal{HS}(\mathcal{H})$.

Let $\mathcal{H}$ be a complex inner product space, and let $HSOp(\mathcal{H})$ denote the space of Hilbert-Schmidt operators on $\mathcal{H}$. For any element $A$ in the opposite algebra of bounded linear operators $\mathcal{B}(\mathcal{H})^{\text{op}}$, the $\star$-algebra homomorphism $\text{rightMulHSStarAlgHom}$ maps $A$ to the right multiplication operator $R_{A^{\text{unop}}}$. Specifically, $\text{rightMulHSStarAlgHom}(A)$ is the continuous linear map on $HSOp(\mathcal{H})$ defined by $T \mapsto T A^{\text{unop}}$, where $A^{\text{unop}}$ is the operator in $\mathcal{B}(\mathcal{H})$ corresponding to $A$.

Let $\mathcal{B}(\mathcal{H})$ denote the algebra of bounded linear operators on a complex inner product space $\mathcal{H}$, and let $\mathcal{B}(\mathcal{H})^\text{op}$ be its opposite algebra. This definition is the $\mathbb{R}$-$\star$-algebra homomorphism from $\mathcal{B}(\mathcal{H})$ to $\mathcal{B}(\mathcal{H})^\text{op}$ that maps an operator $A$ to $(A^*)^\text{op}$, where $A^*$ is the adjoint of $A$. For any self-adjoint operator $A$, the map simply sends $A$ to its corresponding element $A^\text{op}$ in the opposite algebra.

Let $\mathcal{H}$ be a complex inner product space and let $\mathcal{B}(\mathcal{H})$ denote the space of bounded linear operators on $\mathcal{H}$. For any operator $A \in \mathcal{B}(\mathcal{H})$, if $A$ is self-adjoint, then its corresponding element $A^{op}$ in the opposite algebra $\mathcal{B}(\mathcal{H})^{op}$ is also self-adjoint.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathcal{H}$. Suppose $\mathcal{B}(\mathcal{H})$ is nontrivial and admits a continuous functional calculus for self-adjoint operators. For any self-adjoint operator $A \in \mathcal{B}(\mathcal{H})$, let $A^{\text{op}}$ denote its representation in the opposite algebra $\mathcal{B}(\mathcal{H})^{\text{op}}$. Then the spectrum of $A^{\text{op}}$ is equal to the spectrum of $A$, that is, $$\sigma(A^{\text{op}}) = \sigma(A).$$

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathcal{H}$. Suppose that $\mathcal{B}(\mathcal{H})$ is nontrivial and that both $\mathcal{B}(\mathcal{H})$ and its opposite algebra $\mathcal{B}(\mathcal{H})^{\text{op}}$ admit a continuous functional calculus for self-adjoint operators. For any self-adjoint operator $A \in \mathcal{B}(\mathcal{H})$ and any real-valued function $f: \mathbb{R} \to \mathbb{R}$ that is continuous on the spectrum $\sigma(A)$, the functional calculus applied to $A$ in the opposite algebra satisfies: $$f(A^{\text{op}}) = (f(A))^{\text{op}},$$ where $A^{\text{op}}$ denotes the element $A$ viewed as an element of $\mathcal{B}(\mathcal{H})^{\text{op}}$.

Let $\mathcal{H}$ be a complex Hilbert space and $HSOp(\mathcal{H})$ be the space of Hilbert-Schmidt operators on $\mathcal{H}$. For any self-adjoint bounded linear operator $A \in \mathcal{B}(\mathcal{H})$ and any real-valued function $f: \mathbb{R} \to \mathbb{R}$ that is continuous on the spectrum $\sigma(A)$, let $L_A: HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ denote the left multiplication operator defined by $L_A(T) = AT$. Then the continuous functional calculus satisfies: $$L_{f(A)} = f(L_A),$$ where $f(A)$ is the functional calculus of $A$ in $\mathcal{B}(\mathcal{H})$, and $f(L_A)$ is the functional calculus of the operator $L_A$ acting on the Hilbert space $HSOp(\mathcal{H})$.

Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ be the algebra of bounded linear operators on $\mathcal{H}$. Let $HSOp(\mathcal{H})$ be the space of Hilbert-Schmidt operators on $\mathcal{H}$. For any $B \in \mathcal{B}(\mathcal{H})$, let $R_B: HSOp(\mathcal{H}) \to HSOp(\mathcal{H})$ be the continuous linear map defined by right multiplication, $R_B(T) = TB$. Suppose that $\mathcal{B}(\mathcal{H})$ is nontrivial and that both $\mathcal{B}(\mathcal{H})$ and its opposite algebra $\mathcal{B}(\mathcal{H})^{\text{op}}$ admit a continuous functional calculus for self-adjoint operators. For any self-adjoint operator $A \in \mathcal{B}(\mathcal{H})$ and any real-valued function $f: \mathbb{R} \to \mathbb{R}$ continuous on the spectrum $\sigma(A)$, it holds that: $$R_{f(A)} = f(R_A),$$ where $f(A)$ is the functional calculus of $A$ in $\mathcal{B}(\mathcal{H})$, and $f(R_A)$ is the functional calculus of the operator $R_A$ acting on the Hilbert space $HSOp(\mathcal{H})$.