Definitions for Reflection positive one-parameter groups and dilations

QAlgorithms.Papers.Independent.Intakethird100.P92.CompressionMultiplicative

Let $E$ be a complex Hilbert space, let $E_+\subseteq E$ be a closed complex linear subspace, and let $\theta$ be an involutive complex linear isometric bijection satisfying $\operatorname{Re}\langle\theta v,v\rangle\geq0$ for all $v\in E_+$. Let $(U_t)_{t\in\mathbb R}$ be a group of complex linear isometric bijections with identity $U_0$, continuous orbit maps, $U_t(E_+)\subseteq E_+$ for $t\geq0$, and $\theta U_t\theta=U_{-t}$. Write $Z=\{v\in E_+:\theta v=v\}$, and define $C_t:Z\to Z$ by $C_t(v)=P_Z(U_tv)$, where $P_Z$ is the orthogonal projection onto $Z$. This predicate means precisely that

$$C_{s+t}=C_s\circ C_t\qquad\text{for every pair of real numbers }s>0,\ t>0.$$

Equivalently, $P_Z(U_{s+t}v)=P_Z(U_s(P_Z(U_tv)))$ for all such $s,t$ and every $v\in Z$, using the inclusion of $Z$ into $E$ wherever needed. The quantified times are strictly positive.

QAlgorithms.Papers.Independent.Intakethird100.P92.IsCyclic

For the ambient complex inner product space $E$, let $(U_t)_{t\in\mathbb R}$ be a family of complex-linear isometric equivalences $E\to E$ satisfying $U_0x=x$ and $U_{s+t}x=U_s(U_tx)$ for every $s,t\in\mathbb R$ and $x\in E$. For every complex subspace $F\subseteq E$, the declaration defines cyclicity of $F$ to mean that the topological closure in $E$ of $\operatorname{span}_{\mathbb C}\{U_tv:t\in\mathbb R,\ v\in F\}$ equals all of $E$. No continuity of the family in $t$, closedness of $F$, or nonzero assumption on $F$ is required; $F=\{0\}$ satisfies this condition exactly when $E=\{0\}$.

QAlgorithms.Papers.Independent.Intakethird100.P92.IsHermitianForm

Let $K$ be a field equipped with a real-or-complex scalar structure, with conjugation denoted by $a\mapsto\overline a$, let $V$ be an additive commutative group equipped with a $K$-module structure, and let $\beta:V\times V\to K$ be any function. This predicate is the conjunction of the following identities, for every $x,y,z\in V$ and every $a\in K$:

$$\begin{aligned} \beta(x+y,z)&=\beta(x,z)+\beta(y,z),& \beta(ax,y)&=a\,\beta(x,y),\\ \beta(x,y+z)&=\beta(x,y)+\beta(x,z),& \beta(x,ay)&=\overline a\,\beta(x,y),\\ \overline{\beta(x,y)}&=\beta(y,x). \end{aligned}$$

Thus the specified convention is linearity in the first argument and conjugate linearity in the second, together with conjugate symmetry. No positivity, nondegeneracy, topology, or dimension hypothesis is part of this predicate, and the zero module is included.

QAlgorithms.Papers.Independent.Intakethird100.P92.IsPositiveKernel

Let $K$ be a field equipped with a real-or-complex scalar structure, with its specified conjugation and embedding of $\mathbb R$, let $X$ be any type, and let $k:X\times X\to K$ be any function. This predicate says that for every natural number $n$, every indexed family $x_0,\ldots,x_{n-1}\in X$, and every family of coefficients $c_0,\ldots,c_{n-1}\in K$, there exists a real number $a\geq0$ such that

$$\sum_{i=0}^{n-1}\sum_{j=0}^{n-1} \overline{c_i}\,k(x_i,x_j)\,c_j = a,$$

where the right side is embedded into $K$. The points may repeat and the coefficients may vanish. The quantifier includes $n=0$, when the sum is empty and equals zero, so $a=0$ supplies the required witness. The type $X$ may also be empty; then no indexed families exist for $n>0$.

QAlgorithms.Papers.Independent.Intakethird100.P92.IsTame

For any measure $\rho$ on the nonnegative real numbers, $\operatorname{IsTame}(\rho)$ means that the extended nonnegative integral

$$\int_{[0,\infty)}\frac{1}{1+y^2}\,d\rho(y)$$

is strictly less than $+\infty$, with $y$ regarded as an extended nonnegative real in the integrand. No assumption that $\rho$ is nonzero or has finite total mass is included. The integrand is $1$ at $y=0$, and the zero measure satisfies this condition.

QAlgorithms.Papers.Independent.Intakethird100.P92.IsTemperedMeasure

For any type $X$ equipped with a measurable-space structure and a specified real-valued norm operation $x\mapsto\|x\|$, and any measure $\mu$ on $X$, $\operatorname{IsTemperedMeasure}(\mu)$ means that there exists a natural number $N$, including possibly $N=0$, such that

$$\int_X \left(\frac{1}{1+\operatorname{ofReal}(\|x\|^2)}\right)^N \,d\mu(x)<+\infty.$$

This is an extended nonnegative integral, and $\operatorname{ofReal}(r)=\max(r,0)$ regarded as an extended nonnegative real. Only the norm operation is assumed here; no norm axioms or compatibility with the measurable structure are required by the declaration. For $N=0$, the integrand is identically $1$, so this choice expresses finiteness of the total mass. The definition also applies to empty $X$ and to the zero measure.

QAlgorithms.Papers.Independent.Intakethird100.P92.MarkovCondition

Let $E$ be a complex Hilbert space, let $E_+\subseteq E$ be a closed complex linear subspace, and let $\theta:E\to E$ be an involutive complex linear isometric bijection satisfying $\operatorname{Re}\langle\theta v,v\rangle\geq0$ for every $v\in E_+$. Set $Z=\{v\in E_+:\theta v=v\}$. For a subspace $F\subseteq E$, write $\Pi_F:E\to E$ for its ambient-valued orthogonal projection if $F$ is closed and for the zero operator otherwise. This predicate is the operator equality

$$\Pi_{E_+}\circ\Pi_Z\circ\Pi_{\theta(E_+)} = \Pi_{E_+}\circ\Pi_{\theta(E_+)}.$$

In this formula all three target subspaces are closed, so the maps used are their orthogonal projections. Equivalently, the two sides have equal values on every vector of $E$.

QAlgorithms.Papers.Independent.Intakethird100.P92.OS

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, where $P$ is a closed complex linear subspace and $\theta$ is an involutive bijective complex linear isometry satisfying $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for every $v\in P$, $\operatorname{OS}(r)$ is defined to be the uniform-space completion of the records storing vectors of $P$, using the uniform structure induced by the pseudometric $d(x,y)=\sqrt{\operatorname{Re}\langle\theta(v_x-v_y),v_x-v_y\rangle_E}$. This completion identifies records at distance zero and completes the resulting separated space.

QAlgorithms.Papers.Independent.Intakethird100.P92.OSPreSpace

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, where $P\subseteq E$ is a closed complex linear subspace and $\theta$ is a bijective complex linear isometry satisfying $\theta^2=\mathrm{id}$ and $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for every $v\in P$, the type $\operatorname{OSPreSpace}(r)$ consists of records containing exactly one value $v\in P$. Thus each element stores a vector of $E$ together with its membership in $P$, with no additional condition on that vector.

QAlgorithms.Papers.Independent.Intakethird100.P92.RealUnitaryAction.StronglyContinuous

For the ambient complex inner product space $E$, let $(U_t)_{t\in\mathbb R}$ be a family of complex-linear isometric equivalences $E\to E$ satisfying $U_0x=x$ and $U_{s+t}x=U_s(U_tx)$ for every $s,t\in\mathbb R$ and $x\in E$. The declaration defines strong continuity of this family to mean that, for every $x\in E$, the function $\mathbb R\to E$ given by $t\mapsto U_tx$ is continuous.

QAlgorithms.Papers.Independent.Intakethird100.P92.RealUnitaryAction

For the ambient complex inner product space $E$, this structure consists of a family $(U_t)_{t\in\mathbb R}$ of complex-linear isometric equivalences $E\to E$, together with the identities $U_0x=x$ for every $x\in E$ and $U_{s+t}x=U_s(U_tx)$ for every $s,t\in\mathbb R$ and every $x\in E$. The parameters range over all real numbers, including zero and negative values. The structure includes no requirement that the family depend continuously on its real parameter.

QAlgorithms.Papers.Independent.Intakethird100.P92.ReflectionData

For the ambient complex inner product space $E$, reflection data consist of a complex linear subspace $P\subseteq E$, a proof that $P$ is closed, and a bijective complex linear isometry $\theta:E\to E$, together with the requirements that $\theta(\theta x)=x$ for every $x\in E$ and $\operatorname{Re}\langle\theta x,x\rangle_E\geq 0$ for every $x\in P$. The positivity requirement is non-strict and does not require $\langle\theta x,x\rangle_E=0$ to imply $x=0$. The zero subspace is permitted.

QAlgorithms.Papers.Independent.Intakethird100.P92.ReflectionGroup

Let $E$ be a complex inner product space, and let $r$ consist of a closed complex linear subspace $H_+\subseteq E$ and a complex linear surjective isometry $\theta:E\to E$ such that $\theta^2=I$ and $\operatorname{Re}\langle\theta v,v\rangle\geq0$ for every $v\in H_+$. A reflection group for $r$ consists of complex linear surjective isometries $U_t:E\to E$, indexed by all $t\in\mathbb R$, such that $U_0x=x$ and $U_{s+t}x=U_s(U_tx)$ for every $s,t\in\mathbb R$ and $x\in E$; for each $x\in E$, the map $t\mapsto U_tx$ is continuous; $U_tx\in H_+$ whenever $t\geq0$ and $x\in H_+$, including $t=0$; and $\theta(U_t(\theta x))=U_{-t}x$ for every $t\in\mathbb R$ and $x\in E$.

QAlgorithms.Papers.Independent.Intakethird100.P92.domainQ

Let $E$ be a complex inner product space, and let $r$ consist of a closed complex linear subspace $H_+$ and a complex linear surjective isometry $\theta$ satisfying $\theta^2=I$ and $\operatorname{Re}\langle\theta v,v\rangle\geq0$ for every $v\in H_+$. Let $\mathcal H_r$ be the separated completion of $H_+$ with form $[v,w]=\langle\theta v,w\rangle$ and seminorm $\sqrt{\operatorname{Re}\langle\theta v,v\rangle}$, with canonical complex linear map $q:H_+\to\mathcal H_r$. For every complex linear subspace $D$ of $H_+$ and every $v\in D$, the domain map returns $q(v)$ regarded as an element of the image subspace $q(D)\subseteq\mathcal H_r$; its membership in that subspace is witnessed by $v$ itself.

QAlgorithms.Papers.Independent.Intakethird100.P92.logReciprocal

For every nonnegative real number $y$, $\operatorname{logReciprocal}(y)$ is the extended nonnegative real defined by

$$\operatorname{logReciprocal}(y)= \begin{cases} +\infty,&y=0,\\ \max\{\log(1/y),0\},&y>0, \end{cases}$$

where the finite value in the second branch is embedded into the extended nonnegative reals. In particular, it equals $\log(1/y)$ for $0<y<1$, and it equals zero for $y\geq1$.

QAlgorithms.Papers.Independent.Intakethird100.P92.orbitSpan

For the ambient complex inner product space $E$, let $(U_t)_{t\in\mathbb R}$ be a family of complex-linear isometric equivalences $E\to E$ satisfying $U_0x=x$ and $U_{s+t}x=U_s(U_tx)$ for every $s,t\in\mathbb R$ and $x\in E$. For every complex subspace $F\subseteq E$ and every set $S\subseteq\mathbb R$, the declaration defines the complex subspace $\operatorname{span}_{\mathbb C}\{U_tv:t\in S,\ v\in F\}$. This is the algebraic span, consisting of finite complex linear combinations of the indicated vectors. The set $S$ may be empty, in which case the resulting subspace is $\{0\}$.

QAlgorithms.Papers.Independent.Intakethird100.P92.osCompleteSpace

For an ambient complex inner product space $E$, let $r$ consist of a closed complex subspace $P\subseteq E$ and a complex-linear isometric equivalence $\theta:E\to E$ such that $\theta^2x=x$ for every $x\in E$ and $\operatorname{Re}\langle\theta x,x\rangle_E\geq0$ for every $x\in P$. Let $H_r$ be the uniform completion of a copy of $P$ carrying the seminorm $x\mapsto\sqrt{\operatorname{Re}\langle\theta x,x\rangle_E}$, with points at seminorm distance zero identified. This instance supplies completeness of $H_r$: every Cauchy filter on $H_r$ converges.

QAlgorithms.Papers.Independent.Intakethird100.P92.osCore

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, where $P$ is a closed complex linear subspace and $\theta$ is an involutive bijective complex linear isometry satisfying $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for all $v\in P$, this defines a pre-inner-product core on records storing vectors of $P$ by $B(x,y)=\langle\theta v_x,v_y\rangle_E$, where $v_x,v_y$ are the stored vectors. It supplies the identities $\overline{B(y,x)}=B(x,y)$, $B(x+y,z)=B(x,z)+B(y,z)$, and $B(cx,y)=\overline c\,B(x,y)$ for all records $x,y,z$ and all $c\in\mathbb C$, together with $\operatorname{Re}B(x,x)\geq0$ for every $x$. No nondegeneracy requirement is included.

QAlgorithms.Papers.Independent.Intakethird100.P92.osExtension

For an ambient complex inner product space $E$, let $r$ consist of a closed complex subspace $P\subseteq E$ and a complex-linear isometric equivalence $\theta:E\to E$ satisfying $\theta^2x=x$ for every $x\in E$ and $\operatorname{Re}\langle\theta x,x\rangle_E\geq0$ for every $x\in P$. Let $H_r$ be the uniform completion of a copy of $P$ with seminorm $x\mapsto\sqrt{\operatorname{Re}\langle\theta x,x\rangle_E}$, and let $q_r:P\to H_r$ be the canonical complex-linear map. For every complex-linear map $T:P\to P$, the declaration defines a continuous complex-linear map $H_r\to H_r$ as follows: if there exists a continuous complex-linear map $A:H_r\to H_r$ satisfying $A(q_r(x))=q_r(Tx)$ for every $x\in P$, it selects such an $A$ by classical choice; if no such $A$ exists, it returns the zero operator. No continuity assumption on $T$ or existence assumption on $A$ is imposed.

QAlgorithms.Papers.Independent.Intakethird100.P92.osInnerProductSpace

For an ambient complex inner product space $E$, let $r$ consist of a closed complex subspace $P\subseteq E$ and a complex-linear isometric equivalence $\theta:E\to E$ satisfying $\theta^2x=x$ for every $x\in E$ and $\operatorname{Re}\langle\theta x,x\rangle_E\geq0$ for every $x\in P$. Give a copy of $P$ the form $[x,y]_r=\langle\theta x,y\rangle_E$ and the seminorm $\sqrt{\operatorname{Re}[x,x]_r}$. This instance equips its uniform completion $H_r$ with the inherited complex inner product space structure. In particular, if $q_r(x)$ denotes the canonical image of $x\in P$, then $\langle q_r(x),q_r(y)\rangle_{H_r}=\langle\theta x,y\rangle_E$. The completion identifies vectors whose difference has seminorm zero, so the canonical map need not be injective.

QAlgorithms.Papers.Independent.Intakethird100.P92.osNormedAddCommGroup

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, where $P$ is a closed complex linear subspace and $\theta$ is an involutive bijective complex linear isometry satisfying $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for every $v\in P$, let $\operatorname{OS}(r)$ be the uniform completion of the records storing vectors of $P$ with seminorm $\|x\|=\sqrt{\operatorname{Re}\langle\theta v_x,v_x\rangle_E}$. This declaration equips $\operatorname{OS}(r)$ with the normed additive commutative group structure inherited from that completion. Addition, negation, and the norm extend those of the original seminormed group, and in the completion a vector has norm zero exactly when it is zero.

QAlgorithms.Papers.Independent.Intakethird100.P92.osNormedSpace

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, where $P$ is a closed complex linear subspace and $\theta$ is an involutive bijective complex linear isometry satisfying $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for every $v\in P$, let $\operatorname{OS}(r)$ be the uniform completion of the records storing vectors of $P$, whose scalar multiplication acts on stored vectors and whose seminorm is $\|x\|=\sqrt{\operatorname{Re}\langle\theta v_x,v_x\rangle_E}$. This declaration equips $\operatorname{OS}(r)$ with the complex normed-space structure inherited from that completion: complex scalar multiplication extends the original scalar multiplication and satisfies $\|cz\|=|c|\|z\|$ for every $c\in\mathbb C$ and every $z\in\operatorname{OS}(r)$.

QAlgorithms.Papers.Independent.Intakethird100.P92.osNull

For an ambient complex inner product space $E$, let $r$ consist of a closed complex subspace $P\subseteq E$ and a complex-linear isometric equivalence $\theta:E\to E$ satisfying $\theta^2x=x$ for every $x\in E$ and $\operatorname{Re}\langle\theta x,x\rangle_E\geq0$ for every $x\in P$. Let $H_r$ be the uniform completion of a copy of $P$ with seminorm $x\mapsto\sqrt{\operatorname{Re}\langle\theta x,x\rangle_E}$, and let $q_r:P\to H_r$ send each vector to its canonical image in that completion. The declaration defines the complex subspace of $E$ obtained by including $\ker q_r$ into $E$, namely $\{x\in E:x\in P\text{ and }q_r(x)=0\}$. This includes zero and may include nonzero vectors of $E$.

QAlgorithms.Papers.Independent.Intakethird100.P92.osPreAddCommGroup

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, where $P$ is a closed complex linear subspace and $\theta$ is an involutive bijective complex linear isometry with $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for all $v\in P$, this equips the records storing vectors of $P$ with an additive commutative group structure by transporting the structure on $P$ through the bijection that extracts the stored vector. Explicitly, zero stores $0$, the sum of records storing $v,w$ stores $v+w$, and the negative of a record storing $v$ stores $-v$.

QAlgorithms.Papers.Independent.Intakethird100.P92.osPreEquiv

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, with $P$ closed, $\theta$ a bijective complex linear isometry satisfying $\theta^2=\mathrm{id}$, and $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for every $v\in P$, this defines a bijection between the records $\operatorname{OSPreSpace}(r)$ that store a single vector of $P$ and $P$ itself. The forward map extracts the stored vector, and the inverse places a vector into such a record; both compositions are the identity.

QAlgorithms.Papers.Independent.Intakethird100.P92.osPreInnerProductSpace

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, where $P$ is a closed complex linear subspace and $\theta$ is an involutive bijective complex linear isometry satisfying $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for every $v\in P$, this equips the records storing vectors of $P$ with the complex inner-product structure obtained from the form $\langle x,y\rangle_r=\langle\theta v_x,v_y\rangle_E$. This form is conjugate symmetric, conjugate linear in the first variable, linear in the second, and has nonnegative real part on the diagonal; its associated seminorm satisfies $\|x\|^2=\operatorname{Re}\langle\theta v_x,v_x\rangle_E$. The underlying space is seminormed, and zero seminorm is not required here to imply $x=0$.

QAlgorithms.Papers.Independent.Intakethird100.P92.osPreModule

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, where $P$ is a closed complex linear subspace and $\theta$ is an involutive bijective complex linear isometry satisfying $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for every $v\in P$, this equips the records storing vectors of $P$ with a complex module structure transported from $P$. For every $c\in\mathbb C$, scalar multiplication by $c$ sends the record storing $v$ to the record storing $cv$, with addition and zero also transported from $P$.

QAlgorithms.Papers.Independent.Intakethird100.P92.osPreNormedSpace

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, where $P$ is a closed complex linear subspace and $\theta$ is an involutive bijective complex linear isometry satisfying $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for every $v\in P$, this supplies the complex normed-space compatibility structure on records storing vectors of $P$, using scalar multiplication on their stored vectors and the seminorm $\|x\|=\sqrt{\operatorname{Re}\langle\theta v_x,v_x\rangle_E}$ induced by the form $\langle\theta v_x,v_y\rangle_E$. In particular, $\|cx\|=|c|\|x\|$ for every $c\in\mathbb C$ and every record $x$. The underlying additive structure is seminormed, so this declaration does not require zero seminorm to imply that the stored vector is zero.

QAlgorithms.Papers.Independent.Intakethird100.P92.osPreSeminormedAddCommGroup

For an ambient complex inner product space $E$ and reflection data $r=(P,\theta)$, where $P$ is a closed complex linear subspace and $\theta$ is an involutive bijective complex linear isometry satisfying $\operatorname{Re}\langle\theta v,v\rangle_E\geq0$ for every $v\in P$, this equips the additive commutative group of records storing vectors of $P$ with the seminormed structure induced by $B(x,y)=\langle\theta v_x,v_y\rangle_E$. Its seminorm is $\|x\|=\sqrt{\operatorname{Re}\langle\theta v_x,v_x\rangle_E}$, and its distance is $d(x,y)=\sqrt{\operatorname{Re}\langle\theta(v_x-v_y),v_x-v_y\rangle_E}$. Distinct records may have distance zero.

QAlgorithms.Papers.Independent.Intakethird100.P92.osQ

For an ambient complex inner product space $E$, let $r$ consist of a closed complex subspace $P\subseteq E$ and a complex-linear isometric equivalence $\theta:E\to E$ satisfying $\theta^2x=x$ for every $x\in E$ and $\operatorname{Re}\langle\theta x,x\rangle_E\geq0$ for every $x\in P$. Form a copy $P_{\mathrm{pre}}$ of $P$, with its transported complex vector space operations and form $[x,y]_r=\langle\theta x,y\rangle_E$, and let $H_r$ be its uniform completion for the resulting seminorm $\sqrt{\operatorname{Re}[x,x]_r}$. The declaration defines the complex-linear map $q_r:P\to H_r$ by first placing $x$ in its corresponding record in $P_{\mathrm{pre}}$ and then applying the canonical map into the completion. Vectors differing by a vector of seminorm zero have the same image.

QAlgorithms.Papers.Independent.Intakethird100.P92.polynomialValue

For every natural number $d$, every polynomial $p$ with complex coefficients in variables indexed by the $d$-element set $\{0,\ldots,d-1\}$, and every vector $x$ in real Euclidean $d$-space, $\operatorname{polynomialValue}(p,x)$ is the complex number obtained by evaluating each variable indexed by $i$ at the real coordinate $x_i$, embedded into $\mathbb C$:

$$\operatorname{polynomialValue}(p,x) =p(x_0,\ldots,x_{d-1}).$$

The definition includes $d=0$, when there are no variables and evaluation returns the polynomial’s constant value at the unique vector in zero-dimensional Euclidean space.

QAlgorithms.Papers.Independent.Intakethird100.P92.positiveRestriction

Let $E$ be a complex inner product space, and let $r$ consist of a closed complex linear subspace $H_+$ and a complex linear surjective isometry $\theta$ satisfying $\theta^2=I$ and $\operatorname{Re}\langle\theta v,v\rangle\geq0$ for all $v\in H_+$. Let $U_t$, $t\in\mathbb R$, be complex linear surjective isometries satisfying $U_0=I$, $U_{s+t}=U_sU_t$, continuity of every orbit $t\mapsto U_tx$, preservation of $H_+$ for all $t\geq0$, and $\theta U_t\theta=U_{-t}$. For each nonnegative real number $t$, including zero, the positive restriction is the complex linear map $H_+\to H_+$ sending $v$ to $U_tv$; its value belongs to $H_+$ by the assumed preservation property.

QAlgorithms.Papers.Independent.Intakethird100.P92.projectionOn

For every complex linear subspace $F$ of the ambient complex Hilbert space $E$, this defines the continuous complex linear map $\Pi_F:E\to E$ obtained by first mapping $E$ to $F$ by the orthogonal projection when $F$ is closed, or by the zero map when $F$ is not closed, and then including $F$ into $E$. Consequently, $\Pi_F$ is the ambient-valued orthogonal projection onto $F$ in the closed case and the zero operator in the nonclosed case.

QAlgorithms.Papers.Independent.Intakethird100.P92.projectionTo

For a complex linear subspace $F$ of the ambient complex Hilbert space $E$, this defines a continuous complex linear map $P_F:E\to F$ by a case distinction: if $F$ is closed as a subset of $E$, use its resulting completeness to take the orthogonal projection onto $F$, with values regarded as elements of $F$; if $F$ is not closed, take the zero map $E\to F$. Thus the definition applies to every subspace, including the zero subspace and nonclosed subspaces.

QAlgorithms.Papers.Independent.Intakethird100.P92.quantized

Let $E$ be a complex inner product space, with a closed complex linear subspace $H_+$ and a complex linear surjective isometry $\theta$ satisfying $\theta^2=I$ and $\operatorname{Re}\langle\theta v,v\rangle\geq0$ for every $v\in H_+$. Let $\mathcal H_r$ be the separated completion of $H_+$ equipped with the sesquilinear form $[v,w]=\langle\theta v,w\rangle$ and seminorm $\sqrt{\operatorname{Re}\langle\theta v,v\rangle}$, and let $q:H_+\to\mathcal H_r$ be its canonical complex linear map; in particular, vectors at zero seminorm distance are identified. Let $U_t$, $t\in\mathbb R$, be complex linear surjective isometries satisfying $U_0=I$, $U_{s+t}=U_sU_t$, continuity of every orbit, $U_tH_+\subseteq H_+$ for $t\geq0$, and $\theta U_t\theta=U_{-t}$. For each nonnegative real $t$, the quantized operator $Q_t:\mathcal H_r\to\mathcal H_r$ is defined as follows: if there exists a continuous complex linear map $A:\mathcal H_r\to\mathcal H_r$ satisfying $A(qv)=q(U_tv)$ for every $v\in H_+$, then $Q_t$ is a map chosen from those satisfying this condition; if no such map exists, $Q_t$ is the zero operator.

QAlgorithms.Papers.Independent.Intakethird100.P92.quantizedFixed

Let $E$ be a complex inner product space, with a closed complex linear subspace $H_+$ and a complex linear surjective isometry $\theta$ satisfying $\theta^2=I$ and $\operatorname{Re}\langle\theta v,v\rangle\geq0$ for all $v\in H_+$. Write $\mathcal H_r$ for the separated completion of $H_+$ with form $[v,w]=\langle\theta v,w\rangle$ and seminorm $\sqrt{\operatorname{Re}\langle\theta v,v\rangle}$, and $q:H_+\to\mathcal H_r$ for the canonical complex linear map. Let $U_t$, $t\in\mathbb R$, be complex linear surjective isometries with $U_0=I$, $U_{s+t}=U_sU_t$, continuous orbits, preservation of $H_+$ for nonnegative times, and $\theta U_t\theta=U_{-t}$. For $t\geq0$, define $Q_t$ to be a chosen continuous complex linear operator satisfying $Q_t(qv)=q(U_tv)$ for every $v\in H_+$, if such an operator exists, and the zero operator otherwise. The quantized fixed subspace is the complex linear subspace

$$\bigcap_{t>0}\ker(Q_t-I) =\{x\in\mathcal H_r:\ Q_tx=x\text{ for every strictly positive real }t\}.$$

The intersection is indexed by strictly positive times only.

QAlgorithms.Papers.Independent.Intakethird100.P92.quotientImage

Let $E$ be a complex inner product space, and let $r$ consist of a closed complex linear subspace $H_+\subseteq E$ and a complex linear surjective isometry $\theta:E\to E$ satisfying $\theta^2=I$ and $\operatorname{Re}\langle\theta v,v\rangle\geq0$ for all $v\in H_+$. Let $\mathcal H_r$ be the separated completion of $H_+$ for the form $[v,w]=\langle\theta v,w\rangle$ and seminorm $\sqrt{\operatorname{Re}\langle\theta v,v\rangle}$, with canonical complex linear map $q:H_+\to\mathcal H_r$. For any complex linear subspace $D\subseteq E$, its quotient image is the set

$$\{q(v):v\in H_+\text{ and }v\in D\}\subseteq\mathcal H_r.$$

This is the image itself, without taking its closure, and $D$ is not required to be contained in $H_+$.

QAlgorithms.Papers.Independent.Intakethird100.P92.stieltjesDensity

For every natural number $d$, every measure $\rho$ on $[0,\infty)$, and every vector $x$ in real Euclidean $d$-space, the function $\operatorname{stieltjesDensity}(d,\rho,x)$ is the extended nonnegative real

$$f_{d,\rho}(x)= \int_{[0,\infty)} \bigl(y^2+\|x\|^2\bigr)^{-1}\,d\rho(y),$$

where $y$ is embedded into the extended nonnegative reals and $\|x\|^2$ is embedded using $\operatorname{ofReal}(r)=\max(r,0)$. The integral may equal $+\infty$; the inverse is taken in the extended nonnegative reals, so the integrand is $+\infty$ when $y=0$ and $x=0$. The definition includes $d=0$, when the Euclidean space consists only of its zero vector, and it includes the zero measure.

QAlgorithms.Papers.Independent.Intakethird100.P92.stieltjesMeasure

For every natural number $d$ and every measure $\rho$ on $[0,\infty)$, $\operatorname{stieltjesMeasure}(d,\rho)$ is the measure obtained by giving Lebesgue measure on real Euclidean $d$-space the extended nonnegative density

$$f_{d,\rho}(x)= \int_{[0,\infty)}\bigl(y^2+\|x\|^2\bigr)^{-1}\,d\rho(y).$$

Thus, for every measurable set $A$,

$$\operatorname{stieltjesMeasure}(d,\rho)(A) =\int_A f_{d,\rho}(x)\,dx.$$

The arithmetic is in the extended nonnegative reals, with $0^{-1}=+\infty$; neither finiteness of the density nor finiteness of the resulting measure is assumed. The definition includes $d=0$ and the zero measure $\rho$.

QAlgorithms.Papers.Independent.Intakethird100.P92.timeZero

For an ambient complex inner product space $E$, let $r$ consist of a closed complex subspace $P\subseteq E$ and a complex-linear isometric equivalence $\theta:E\to E$ satisfying $\theta^2x=x$ for every $x\in E$ and $\operatorname{Re}\langle\theta x,x\rangle_E\geq0$ for every $x\in P$. The declaration defines the complex subspace $P\cap\ker(\theta-\operatorname{id}_E)$, consisting exactly of those $x\in E$ such that $x\in P$ and $\theta x=x$. This subspace contains zero and is allowed to equal $\{0\}$.

QAlgorithms.Papers.Independent.Intakethird100.P92.timeZeroCompression

Let $E$ be a complex Hilbert space, let $E_+\subseteq E$ be a closed complex linear subspace, and let $\theta$ be an involutive complex linear isometric bijection with $\operatorname{Re}\langle\theta v,v\rangle\geq0$ for every $v\in E_+$. Let $(U_t)_{t\in\mathbb R}$ be complex linear isometric bijections satisfying $U_0=\mathrm{id}$, $U_{s+t}=U_sU_t$, continuity of every orbit map, preservation of $E_+$ for $t\geq0$, and $\theta U_t\theta=U_{-t}$. Set $Z=\{v\in E_+:\theta v=v\}$. For every real $t$, this defines a complex linear map $C_t:Z\to Z$ by

$$C_t(v)=P_Z(U_tv),$$

where $v$ is first included into $E$ and $P_Z:E\to Z$ is the orthogonal projection onto the closed subspace $Z$. The underlying projection convention is to use the zero map for a nonclosed target subspace; here $Z$ is closed. The time $t$ may be negative, zero, or positive.

QAlgorithms.Papers.Independent.Intakethird100.P92.timeZeroQuotient

Let $E$ be a complex inner product space, let $E_+\subseteq E$ be a closed complex linear subspace, and let $\theta:E\to E$ be an involutive complex linear isometric bijection satisfying $\operatorname{Re}\langle\theta v,v\rangle\geq0$ for every $v\in E_+$. Set $Z=\{v\in E_+:\theta v=v\}$, and let $H_r$ be the separated completion of a copy of $E_+$ with the possibly degenerate inner product $\langle\theta v,w\rangle$, with canonical complex linear map $q:E_+\to H_r$. This defines the complex linear map $q_0:Z\to H_r$ by including $Z$ into $E_+$ and then applying $q$; explicitly, $q_0(v)=q(v)$ for every $v\in Z$. The definition itself makes no assertion of surjectivity.

QAlgorithms.Papers.Independent.Intakethird100.P92.toOSPre

For an ambient complex inner product space $E$, let $r$ consist of a closed complex subspace $P\subseteq E$ and a complex-linear isometric equivalence $\theta:E\to E$ satisfying $\theta^2x=x$ for every $x\in E$ and $\operatorname{Re}\langle\theta x,x\rangle_E\geq0$ for every $x\in P$. Let $P_{\mathrm{pre}}$ be the type whose elements are records containing one value in $P$, with addition and complex scalar multiplication transported from $P$ through the map that extracts this value. The declaration defines the complex-linear map $P\to P_{\mathrm{pre}}$ sending each $x\in P$ to the record containing $x$; it is the inverse of the value-extraction linear equivalence.

QAlgorithms.Papers.Independent.Intakethird100.P92.unitaryFixedSubspace

For the ambient complex inner product space $E$, let $(U_t)_{t\in\mathbb R}$ be a family of complex-linear isometric equivalences $E\to E$ satisfying $U_0x=x$ and $U_{s+t}x=U_s(U_tx)$ for every $s,t\in\mathbb R$ and $x\in E$. The declaration defines the complex subspace $\bigcap_{t\in\mathbb R}\ker(U_t-\operatorname{id}_E)$, consisting exactly of those $x\in E$ satisfying $U_tx=x$ for every real $t$, including zero and negative times. It always contains zero and is allowed to be the zero subspace.