On sums and convex combinations of projectors onto convex sets

Introduction

with inner product $\langle\,\cdot\,|\,\mathopen{}\cdot\,\rangle$ and induced norm $\lVert\,\cdot\,\rVert$. Now assume thatFor basic Convex Analysis, we refer the reader to .

In this paper, we analyze carefully the question: When is $\sum_{i\in I}\alpha_{i}P_{C_{i}}$ a projector? This allows us to provide a complete answer to the question “When is the sum of projectors also a projector?” (In view of Proposition 2.4(iii), an affirmative answer to this question requires the sum $\sum_{i\in I}C_{i}$ to be closed. This happens, for instance, when each set is bounded.) It is known that, in the case of linear subspaces, $\sum_{i\in I}P_{C_{i}}$ is a projector onto a closed linear subspace if and only if $(C_{i})_{}$ is pairwise orthogonal; see [14, Theorem 2, p. 46]. This question is also of interest in Quantum Mechanics [17, p. 50]. In 1971, Zarantonello answered this question in the case of convex cones, i.e., if $(C_{i})_{}$ are cones, then $\sum_{i\in I}P_{C_{i}}$ is a projector if and only if $(P_{C_{i}})_{}$ is pairwise orthogonal in the sense that, for every $(i,j)\in I\times I$ with $i\neq j$, we have $(\forall x\in\mathcal{H})\,\langle{P_{C_{i}}x}\,|\,\mathopen{}{P_{C_{j}}x}\rangle=0.$ However, the question remains open in the general convex case. Therefore, one goal of this paper is to provide necessary and sufficient conditions for $\sum_{i\in I}\alpha_{i}P_{C_{i}}$ to be a projector without any further assumption on the sets $(C_{i})_{}$. As a consequence, we answer entirely the question “When is the sum of projectors also a projector?” Our results unify the two aforementioned results and make a connection with the recent work where it was proven that, if the sum of a family of proximity operators is a proximity operator, then every partial sum remains a proximity operator. Interestingly, we shall see that this property is still valid in the class of projectors onto convex cones; in other words, if a finite sum of projectors onto convex cones is a projector, then so are its partial sums. Nevertheless, this result fails outside the world of convex cones. Another goal is to characterize the instances where a convex average of $(P_{C_{i}})_{}$ is again a projector. In striking contrast to a result in 1963 by Moreau , which states that a convex average of proximity operators is always a proximity operator, we shall see in Theorem 4.3 that taking convex combinations does not preserve the class of projectors onto convex sets (see Theorem 4.3 for the rigorous statement). Our main results are summarized as follows:

We provide a new characterization of proximity operators in Theorem 3.1 (for a list of other characterizations, see ). In turn, we derive a new characterization of projectors (Theorem 3.2), which is a pillar of this paper and a variant of [26, Theorem 4.1]. Furthermore, we also partially answer an open question by Zarantonello regarding [26, Theorem 4.1].

Theorem 3.10 characterizes (without any additional assumptions on the underlying sets) when $\sum_{i\in I}\alpha_{i}P_{C_{i}}$ is a projector; Theorem 3.12 concerns the sum $\sum_{i\in I}P_{C_{i}}$.

By specifying our analysis to the case of convex average in Theorem 4.3, we explicitly determine families of closed convex sets that are preserved under taking convex combinations.

We present the partial sum property (see [2, Theorem 4.2]) for projectors onto convex cones in Theorem 5.7, whose proof is based on Theorem 5.3 and [2, Theorem 4.2]. We also recover [26, Theorems 5.3 and 5.5].

where $\nabla{\operatorname{q}}={\operatorname{Id}}$ is the identity operator on $\mathcal{H}$. Let $C$ be a subset of $\mathcal{H}$. Then we denote by $\overline{C}$ the closure of $C$ (with respect to the norm topology on $\mathcal{H}$), by $d_{C}$ its distance function, by ${C}^{\ominus}$ its polar cone, i.e., ${C}^{\ominus}\coloneqq\{{u\in\mathcal{H}}~{}|~{}\mathopen{}{\sup\langle{C}\,|\,\mathopen{}{u}\rangle\leqslant 0}\},$ and by $C^{\perp}$ its orthogonal complement. Next, the indicator and support functions of $C$ are

respectively. Moreover, if $C$ is convex, closed, and nonempty, then the projector associated with $C$ is denoted by $P_{C}$. In turn, we set

Next, the set of convex, lower semicontinuous, and proper functions from $\mathcal{H}$ to $\mathopen{}\left]{-}\infty,{+}\infty\right]$ is $\varGamma_{0}(\mathcal{H})$. The domain of of a function $f\colon\mathcal{H}\to\mathopen{}\left[{-}\infty,{+}\infty\right]$ is $\operatorname{dom}{f}\coloneqq\{{x\in\mathcal{H}}~{}|~{}\mathopen{}{f(x)<+\infty}\}$ with closure $\operatorname{\overline{dom}}f$, its graph is denoted by $\operatorname{gra}f$, its conjugate is denoted by $f^{\ast}$, and its subdifferential is denoted by $\partial f$; furthermore, if $f\in\varGamma_{0}(\mathcal{H}),$ then we denote its proximity operator by $\operatorname{Prox}_{f}$ and its Moreau envelope by $\operatorname{env}f$, i.e., \operatorname{env}f\coloneqq f\mbox{\footnotesize\,\square\,}{\operatorname{q}}=f\mbox{\footnotesize\,\boxdot\,}{\operatorname{q}}, where $\,\square\,$ and $\,\boxdot\,$ denote the infimal convolution and the exact infimal convolution, respectively. Next, let $T\colon\mathcal{H}\to\mathcal{H}$. The range of $T$ is $\operatorname{ran}T$ with closure $\operatorname{\overline{ran}}T$. If $T\in\mathscr{B}(\mathcal{H})$, the space of bounded linear operators on $\mathcal{H}$, then its adjoint is denoted by $T^{\ast}$. Finally, we adopt the convention that empty sums are zero.

Auxiliary results

In this section, we provide various results that will be useful in the sequel. Let us start with a simple identity in $\mathcal{H}$.

$\mathopen{}\left\lVert x-\sum_{i\in I}\alpha_{i}x_{i}\right\rVert^{2}=(1-\alpha)\lVert x\rVert^{2}+\sum_{i\in I}\alpha_{i}\lVert x-x_{i}\rVert^{2}+(\alpha-1)\sum_{i\in I}\alpha_{i}\lVert x_{i}\rVert^{2}-\tfrac{1}{2}\sum_{i\in I}\sum_{j\in I}\alpha_{i}\alpha_{j}\lVert x_{i}-x_{j}\rVert^{2}.$

Suppose that $(\forall i\in I)\,\alpha_{i}=1$. Then $\alpha=\operatorname{card}I$ and

Base case: When $m=1$, by applying [4, Corollary 2.15] to $(x-x_{1},x_{1})$ and noticing that $\alpha=\alpha_{1}$, we obtain

Inductive step: Assume that $m\geqslant 2$ and that the result holds for families containing $m-1$ or fewer elements. Moreover, set $J\coloneqq\{1,\ldots,m-1\}$ and $\beta\coloneqq\sum_{j\in J}\alpha_{j}$. Then, by the base case, we have

Hence, since $\beta+\alpha_{m}=\alpha$, we infer from the induction hypothesis that

(ii): Since $(\forall i\in I)\,\alpha_{i}=1$, we have $\alpha=\operatorname{card}I$, and thus

and hence 9 holds. Consequently, 10 follows from (i) and 9. ∎

We shall need the following identities involving convex cones.

Let $K$ and $S$ be nonempty closed convex cones in $\mathcal{H}$. Then the following hold:

$(\forall x\in\mathcal{H})\,\lVert P_{K}x\rVert^{2}=\langle{x}\,|\,\mathopen{}{P_{K}x}\rangle$.

$(\forall x\in\mathcal{H})\,\langle{P_{{K}^{\ominus}}x}\,|\,\mathopen{}{P_{{S}^{\ominus}}x}\rangle+\lVert P_{K}x\rVert^{2}+\lVert P_{S}x\rVert^{2}=\lVert x\rVert^{2}+\langle{P_{K}x}\,|\,\mathopen{}{P_{S}x}\rangle.$

Take $x\in\mathcal{H}$. (i): We derive from [4, Theorem 6.30(i)&(ii)] that $\lVert P_{K}x\rVert^{2}=\langle{P_{K}x}\,|\,\mathopen{}{P_{K}x}\rangle=\langle{x-P_{{K}^{\ominus}}x}\,|\,\mathopen{}{P_{K}x}\rangle=\langle{x}\,|\,\mathopen{}{P_{K}x}\rangle,$ as claimed. (ii): The Moreau conical decomposition () and (i) give

Let $C$ be a nonempty closed convex subset of $\mathcal{H}$. Then the following hold:

$P_{C}$ is $3^{\ast}$ monotoneA monotone operator $A\colon\mathcal{H}\to 2^{\mathcal{H}}$ is $3^{\ast}$ monotone if $(\forall(x,u)\in\operatorname{dom}{A}\times\operatorname{ran}{A})\,\inf_{(y,v)\in\operatorname{gra}{A}}\langle{x-y}\,|\,\mathopen{}{u-v}\rangle>-\infty.$.

(i): See [4, Example 20.32]. (ii): Because $P_{C}$ is firmly nonexpansive by [4, Proposition 4.16], the conclusion follows from [4, Example 25.20(ii)]. ∎

In the finite-dimensional case, Proposition 2.4(ii) can also be deduced from [7, Theorem 3.15]. Furthermore, let us point out that Proposition 2.4(iii) generalizes Zarantonello’s [26, Theorem 5.4].

Suppose that $(\forall i\in I)\,\alpha_{i}\geqslant 0$. Then $\operatorname{\overline{ran}}{\sum_{i\in I}\alpha_{i}P_{C_{i}}}=\overline{\sum_{i\in I}\alpha_{i}C_{i}}.$

Suppose that $(\forall i\in I)\,\alpha_{i}\geqslant 0$ and that there exists a closed convex set $C$ such that $\sum_{i\in I}\alpha_{i}P_{C_{i}}=P_{C}$. Then $\sum_{i\in I}\alpha_{i}C_{i}$ is closed and $C=\sum_{i\in I}\alpha_{i}C_{i}$.

(i): Let $x$ be in $\mathcal{H}$. Apply Lemma 2.1(i) to $(x,(P_{C_{i}}x)_{},(\alpha_{i})_{})$ and notice that $(\forall i\in I)\,\lVert x-P_{C_{i}}x\rVert=d_{C_{i}}(x)$.

(ii): Because the operators $(P_{C_{i}})_{}$ are $3^{\ast}$ monotone by Fact 2.3(ii) and because $(\forall i\in I)\,\operatorname{dom}P_{C_{i}}=\mathcal{H}$, we derive from [9, Lemma 3.1(ii)] that $\operatorname{\overline{ran}}\sum_{i\in I}\alpha_{i}P_{C_{i}}=\overline{\sum_{i\in I}\alpha_{i}\operatorname{ran}P_{C_{i}}}=\overline{\sum_{i\in I}\alpha_{i}C_{i}},$ as desired.

(iii): It follows from (ii) and our assumption that

Thus, we conclude that $\sum_{i\in I}\alpha_{i}C_{i}=C$ and that $\sum_{i\in I}\alpha_{i}C_{i}$ is closed. ∎

Proposition 2.4(ii)&(iii) may fail if $(\exists i\in I)~{}\alpha_{i}<0$. Indeed, in the setting of Proposition 2.4, suppose that $I=\mathopen{}\left\{1,2\right\}$, that $C_{1}=C_{2}$, and that $\alpha_{1}=-\alpha_{2}=1$. Then $\alpha_{1}P_{C_{1}}+\alpha_{2}P_{C_{2}}=0=P_{\mathopen{}\left\{0\right\}}$, but $\alpha_{1}C_{1}+\alpha_{2}C_{2}=C_{1}-C_{1}\neq\mathopen{}\left\{0\right\}$ if $C_{1}$ is not a singleton.

The following is a variant of [27, Lemma 6.1]. We provide a proof for completeness.

By Lemma 2.6 and our assumption, $\nabla f(0)=0$, which implies that

Recall from [18, pp. 89–90] that, if $f\colon\mathcal{H}\to\mathopen{}\left]-\infty,+\infty\right]$, then the Fréchet subdifferential of $f$ is

Main results

Let $\varphi\in\varGamma_{0}(\mathcal{H})$, let $T\colon\mathcal{H}\to\mathcal{H}$, and set $f\coloneqq\varphi\circ T+{\operatorname{q}}\circ({{\operatorname{Id}}}-T)$. Then the following are equivalent:

$T$ is monotone, $\operatorname{gra}(\varphi+\iota_{\operatorname{ran}T})$ is a dense subset of $\operatorname{gra}\varphi$, and $f$ is Gâteaux differentiable on $\mathcal{H}$ with $\nabla f={\operatorname{Id}}-T$.

Furthermore, if (i) or (ii) holds, then $f=\operatorname{env}\varphi$ and $f$ is Fréchet differentiable on $\mathcal{H}$.

“(i)$\Rightarrow$(ii)”: First, by [4, Example 20.30], $T=\operatorname{Prox}_{\varphi}$ is monotone. Next, since $\varphi\in\varGamma_{0}(\mathcal{H})$ and $T=\operatorname{Prox}_{\varphi}$, [4, Eq. (24.3)] gives $\operatorname{ran}T=\operatorname{dom}\partial\varphi$, and hence, according to [4, Proposition 16.38], it follows that $\operatorname{gra}(\varphi+\iota_{\operatorname{ran}T})$ is a dense subset of $\operatorname{gra}\varphi$. Finally, in view of [4, Remark 12.24], we see that $f=\varphi\circ T+{\operatorname{q}}\circ({\operatorname{Id}}-T)=\varphi\circ{\operatorname{Prox}_{\varphi}}+{\operatorname{q}}\circ({\operatorname{Id}}-{\operatorname{Prox}_{\varphi}})=\operatorname{env}\varphi$, and [4, Proposition 12.30] thus entails that $f$ is Fréchet (thus Gâteaux) differentiable on $\mathcal{H}$ with $\nabla f={\operatorname{Id}}-{\operatorname{Prox}_{\varphi}}={\operatorname{Id}}-T$.

“(i)$\Leftarrow$(ii)”: Set $g\coloneqq{\operatorname{q}}-f$. Then, on the one hand, because ${\operatorname{q}}$ and $f$ are Gâteaux differentiable, so is $g$. On the other hand, since $\nabla{\operatorname{q}}={\operatorname{Id}}$ and $\nabla f={\operatorname{Id}}-T$, we infer that $\nabla g=\nabla({\operatorname{q}}-f)=\nabla{\operatorname{q}}-\nabla f=T,$ which is monotone by assumption. Altogether, [4, Proposition 17.7] yields the convexity of $g$. Therefore, since $g$ is Gâteaux differentiable on $\mathcal{H}$, it follows from [4, Proposition 17.48(i)] that $g$ is lower semicontinuous on $\mathcal{H}$. To sum up, we have shown that

Moreover, 24 and [4, Corollary 13.38] yield

In turn, set $h\coloneqq g^{\ast}-{\operatorname{q}}.$ Let us now establish that

Towards this goal, fix $u\in\operatorname{ran}{T}$, say $u=Tx\overset{\lx@cref{creftype~refnum}{e:info-g}}{=}\nabla g(x)$, where $x\in\mathcal{H}$. Then 24, [4, Proposition 17.35], and the very definitions of $g$ and $f$ assert that

Hence, 26 holds. Next, fix $v\in\operatorname{dom}h$, and we shall prove that $\varphi(v)\leqslant h(v)$. Indeed, on the one hand, because $h=g^{\ast}-{\operatorname{q}}$ and $\operatorname{dom}{\operatorname{q}}=\mathcal{H}$, we have $\operatorname{dom}h=\operatorname{dom}g^{\ast}$. On the other hand, due to 24 and [4, Corollary 16.30], $\operatorname{dom}\partial g^{\ast}=\operatorname{dom}(\partial g)^{-1}=\operatorname{ran}\partial g,$ and since $\operatorname{ran}\partial g=\operatorname{ran}\nabla g=\operatorname{ran}T$ thanks to 24 and [4, Proposition 17.31(i)], we deduce that $\operatorname{dom}\partial g^{\ast}=\operatorname{ran}T$. Altogether, because $v\in\operatorname{dom}h=\operatorname{dom}g^{\ast}$, 25 and [4, Proposition 16.38] ensures the existence of a sequence $(v_{n})_{}$ in $\operatorname{dom}\partial g^{\ast}=\operatorname{ran}T$ such that $v_{n}\to v$ and $g^{\ast}(v_{n})\to g^{\ast}(v)$. Therefore, by the definition of $h$, we get $h(v_{n})=g^{\ast}(v_{n})-{\operatorname{q}}(v_{n})\to g^{\ast}(v)-{\operatorname{q}}(v)=h(v)$. However, because $\{v_{n}\}_{}\subseteq\operatorname{ran}T$ and $v_{n}\to v$, the lower semicontinuity of $\varphi$ and 26 imply that $h(v)=\lim h(v_{n})=\lim\varphi(v_{n})\geqslant\varphi(v).$ Hence, we have established that

To this end, let $w\in\operatorname{dom}\varphi$. Then, since $\operatorname{gra}(\varphi+\iota_{\operatorname{ran}T})$ is a dense subset of $\operatorname{gra}\varphi$ by assumption, there exists a sequence $(w_{n})_{}$ in $\operatorname{ran}T$ such that $w_{n}\to w$ and $\varphi(w_{n})\to\varphi(w)$. In turn, since $h$ is lower semicontinuous by 25, we infer from 26 that $\varphi(w)=\lim\varphi(w_{n})=\lim h(w_{n})\geqslant h(w),$ from which 29 follows. Consequently, combining 28 and 29 yields $h=\varphi$. Finally, since $\varphi\in\varGamma_{0}(\mathcal{H})$, it follows from 24, the definition of $h$, the Fenchel–Moreau theorem, and [4, Proposition 24.4] that $\operatorname{Prox}_{\varphi}=\nabla(\varphi+{\operatorname{q}})^{\ast}=\nabla(h+{\operatorname{q}})^{\ast}=\nabla g^{\ast\ast}=\nabla g=T,$ as desired. ∎

In , Zarantonello provided a necessary and sufficient condition in terms of a differential equation for an operator on $\mathcal{H}$ to be a projector. The proof there, however, is not within the scope of Convex Analysis. He also conjectured (see the paragraph after [26, Corollary 2, p. 306]) that the Fréchet differentiability of the operator $P$ in [26, Theorem 4.1] can be replaced by the Gâteaux one. By assuming the monotonicity of $P$ instead of the Lipschitz continuity, we provide below an affirmative answer. The next result, which plays a crucial role in determining whether a sum of projectors is a projector (see Theorem 3.12 below), is a variant of [26, Theorem 4.1] with a proof rooted in Convex Analysis.

Let $T\colon\mathcal{H}\to\mathcal{H},$ and set $f\coloneqq{\operatorname{q}}\circ({\operatorname{Id}}-T)$. Then the following are equivalent:

$T$ is monotone, $f$ is Gâteaux differentiable on $\mathcal{H}$, and $\nabla f={\operatorname{Id}}-T$.

If (i) or (ii) holds, then $\operatorname{ran}{T}$ is closed and convex, $T=P_{\operatorname{ran}{T}}$, and $f=(1/2)d_{\operatorname{ran}{T}}^{2}$ is Fréchet differentiable on $\mathcal{H}$.

Set $\varphi\coloneqq\iota_{\operatorname{\overline{ran}}T}$.

“(i)$\Rightarrow$(ii)”: Suppose that $T=P_{C}$, where $C$ is convex, closed, and nonempty. Then clearly $\operatorname{ran}{T}=\operatorname{ran}{P_{C}}=C$ is closed and convex. This implies that $\varphi=\iota_{\operatorname{ran}T}\in\varGamma_{0}(\mathcal{H})$ and that $T=P_{\operatorname{ran}T}=\operatorname{Prox}_{\iota_{\operatorname{ran}T}}=\operatorname{Prox}_{\varphi}$. In turn, because $f=\varphi\circ T+{\operatorname{q}}\circ({\operatorname{Id}}-T)$ by the definition of $\varphi$, we infer from Theorem 3.1 that (ii) holds and, moreover, $f=\operatorname{env}\varphi=\operatorname{env}\iota_{\operatorname{ran}T}=(1/2)d_{\operatorname{ran}T}^{2}$ is Fréchet differentiable on $\mathcal{H}$.

“(i)$\Leftarrow$(ii)”: We first show that $\operatorname{\overline{ran}}T$ is convex. Indeed, by our assumption, ${\operatorname{q}}-f$ is Gâteaux differentiable on $\mathcal{H}$ with

Thus, since $T$ is monotone, [4, Proposition 17.7] ensures that ${\operatorname{q}}-f$ is convex, and thus, the Gâteaux differentiability of ${\operatorname{q}}-f$ and [4, Proposition 17.48(i)] imply that ${\operatorname{q}}-f\in\varGamma_{0}(\mathcal{H})$. Hence, due to 30 and [4, Proposition 17.31(i)], Moreau’s theorem asserts that $T=\nabla({\operatorname{q}}-f)$ is maximally monotone. Consequently, [4, Corollary 21.14] yields the convexity of $\operatorname{\overline{ran}}T$, as claimed. In turn, on the one hand, this implies that $\varphi=\iota_{\operatorname{\overline{ran}}T}\in\varGamma_{0}(\mathcal{H})$. On the other hand, we deduce from the definition of $\varphi$ that $f=\varphi\circ T+{\operatorname{q}}\circ({\operatorname{Id}}-T)$ and $\operatorname{gra}(\varphi+\iota_{\operatorname{ran}T})=\operatorname{gra}(\iota_{\operatorname{\overline{ran}}T\cap\operatorname{ran}T})=\operatorname{gra}\iota_{\operatorname{ran}T}=\operatorname{ran}T\times\mathopen{}\left\{0\right\}$ is a dense subset of $\operatorname{\overline{ran}}T\times\mathopen{}\left\{0\right\}=\operatorname{gra}\iota_{\operatorname{\overline{ran}}T}=\operatorname{gra}\varphi$. Thus, the implication “(ii)$\Rightarrow$(i)” of Theorem 3.1 and our assumption guarantee that $T=\operatorname{Prox}_{\varphi}=\operatorname{Prox}_{\iota_{\operatorname{\overline{ran}}T}}=P_{\operatorname{\overline{ran}}T}$, which completes the proof. ∎

Consider the implication “(ii)$\Rightarrow$(i)” of Theorem 3.2. If we merely assume that $T$ is defined on a proper open subset $D$ of $\mathcal{H}$, then, although there may exist a closed set $C$ such that $T$ is the restriction to $D$ of the projector onto $C$, the set $C$ may fail to be convex. An example can be constructed as follows. Suppose that $\mathcal{H}\neq\mathopen{}\left\{0\right\}$, and set

i.e., $C$ is the unit sphere of $\mathcal{H}$. Then clearly $C$ is a closed nonconvex set and $T$ is the restriction to $\mathcal{H}\smallsetminus\mathopen{}\left\{0\right\}$ of the set-valued projector $P_{C}$. Thus, in the light of [4, Example 20.12], $T$ is monotone. Next, since $(\forall x\in\mathcal{H}\smallsetminus\mathopen{}\left\{0\right\})\,f(x)=(1/2)\lVert(1-1/\lVert x\rVert)x\rVert^{2}=(1/2)(\lVert x\rVert-1)^{2}={\operatorname{q}}(x)-\lVert x\rVert+1/2,$ we infer that $f$ is Fréchet differentiable on $\mathcal{H}\smallsetminus\mathopen{}\left\{0\right\}$ and

We do not know whether the monotonicity of $T$ can be omitted in Theorem 3.2. Nevertheless, on the one hand, the following remark might be useful in finding counterexamples if one thinks the answer is negative; on the other hand, Proposition 3.6 provides information on the set $\operatorname{Fix}T$ in the absence of monotonicity.

However, because $\nabla f=F$, a direct computation gives

Let $T\colon\mathcal{H}\to\mathcal{H}$, and set $f\coloneqq{\operatorname{q}}\circ({\operatorname{Id}}-T)$. Suppose that $f$ is Fréchet differentiable on $\mathcal{H}$ with $\nabla f={\operatorname{Id}}-T$. Then $\operatorname{Fix}T\neq\varnothing.$

Now let $\varepsilon\in]0,1[$. Since $g$ is bounded below and continuous, Ekeland’s variational principle (see, e.g., [4, Theorem 1.46(iii)]) applied to $g$ and $(\alpha,\beta)=(\varepsilon^{2},\varepsilon)$ yields the existence of $z\in\mathcal{H}$ such that $(\forall x\in\mathcal{H}\smallsetminus\{z\})\,g(z)+\varepsilon d_{\mathopen{}\left\{z\right\}}(z)=g(z)<g(x)+\varepsilon d_{\mathopen{}\left\{z\right\}}(x)$. This guarantees that $z$ is the unique minimizer of $g+\varepsilon d_{\mathopen{}\left\{z\right\}}$. Thus, [18, Proposition 1.114], Lemma 2.8, and 35 imply that

which is absurd since $\varepsilon\in]0,1[$ and $\lVert(z-Tz)/(\lVert z-Tz\rVert)\rVert=1$. ∎

Consider the setting and the assumption of Proposition 3.6.

Zarantonello established in the proof of [26, Theorem 4.1] that, if (in addition to our assumption) $T$ is Lipschitz continuous, then $\operatorname{Fix}{T}\neq\varnothing$. However, we do not need the Lipschitz continuity of $T$ in our proof.

Suppose, in addition, that $\nabla f$ is continuous. Then we obtain an alternative proof as follows. Assume to the contrary that $\operatorname{Fix}T=\varnothing.$ Then $g\coloneqq\sqrt{\,\cdot\,}\circ(2f)$ is continuously Fréchet differentiable on $\mathcal{H}$ (hence continuous) with

Fix $\varepsilon\in]0,1[$. Since $g$ is bounded below and continuous, Ekeland’s variational principle implies that there exists $z\in\mathcal{H}$ such that $(\forall x\in\mathcal{H}\smallsetminus\{z\})\,g(z)+\varepsilon d_{\mathopen{}\left\{z\right\}}(z)=g(z)<g(x)+\varepsilon d_{\mathopen{}\left\{z\right\}}(x)$. Thus, $z$ is a minimizer of $g+\varepsilon d_{\mathopen{}\left\{z\right\}}(z)$. Therefore, because $d_{\mathopen{}\left\{z\right\}}$ is convex, in view of [25, Theorem 3.2.4(iii)&(vi)&(ii)] and [4, Example 16.62], we see that

which contradicts the fact that $\varepsilon\in]0,1[.$

By specializing Theorem 3.2 to positively homogeneous operators on $\mathcal{H}$, we obtain a characterization for projectors onto closed convex cones.

Let $T\colon\mathcal{H}\to\mathcal{H}$ and set $f\coloneqq{\operatorname{q}}\circ T$. Then the following are equivalent:

There exists a nonempty closed convex cone $K$ such that $T=P_{K}$.

$T$ is monotone and positively homogeneous, $f$ is Gâteaux differentiable on $\mathcal{H}$, and $\nabla f=T$.

If (i) or (ii) holds, then $K=\operatorname{ran}{T}$.

“(i)$\Rightarrow$(ii)”: Clearly $\operatorname{ran}{T}=\operatorname{ran}P_{K}=K$. Now, it follows from [4, Example 20.32] that $T=P_{K}$ is monotone. Next, because $K$ is a nonempty closed convex cone, [4, Proposition 29.29] guarantees that $T$ is positively homogeneous. In turn, since $f={\operatorname{q}}\circ T={\operatorname{q}}\circ P_{K}$, [4, Proposition 12.32 and Lemma 2.61(i)] yield the Gâteaux differentiability of $f$ and, moreover, $\nabla f=\nabla({\operatorname{q}}\circ P_{K})=P_{K}=T$, as desired.

“(i)$\Leftarrow$(ii)”: First, since $T$ is positively homogeneous,

In Corollary 3.8, if $T$ is a bounded linear operator, then we recover the following characterization of orthogonal projectors. For an alternative proof, which is based on the orthogonal decomposition $\mathcal{H}=V\oplus V^{\perp}$, where $V$ is a closed linear subspace of $\mathcal{H}$, see, e.g., [24, Theorem 4.29].

Let $L\colon\mathcal{H}\to\mathcal{H}$. Then the following are equivalent:

There exists a closed linear subspace $V$ of $\mathcal{H}$ such that $L=P_{V}$.

$L\in\mathscr{B}(\mathcal{H})$ and $L=L^{\ast}=L^{2}$.

$L\in\mathscr{B}(\mathcal{H})$ and $L=L^{\ast}L$.

If one of (i), (ii) and (iii) holds, then $V=\operatorname{ran}{L}$.

“(i)$\Rightarrow$(ii)”: See, e.g., [4, Corollary 3.24(iii)&(vi)]. Moreover, it is clear that $\operatorname{ran}L=\operatorname{ran}P_{V}=V$.

“(iii)$\Rightarrow$(i)”: On the one hand, because $L\in\mathscr{B}(\mathcal{H})$, we deduce from [4, Example 20.16(ii)] that $L=L^{\ast}L$ is monotone. On the other hand, since $L\in\mathscr{B}(\mathcal{H})$, [4, Example 2.60] and our assumption imply that ${\operatorname{q}}\circ L$ is Fréchet differentiable on $\mathcal{H}$ and $\nabla({\operatorname{q}}\circ L)=L^{\ast}L=L$. Altogether, because $L$ is clearly positively homogeneous, we obtain the conclusion via Corollary 3.8. ∎

Set $T\coloneqq\sum_{i\in I}\alpha_{i}P_{C_{i}}$, set $f\coloneqq{\operatorname{q}}\circ({\operatorname{Id}}-T)$, and define

Now assume that there exists a nonempty closed convex subset $C$ of $\mathcal{H}$ such that $T=P_{C}$. Then, due to Fact 2.3(i), we see that $T$ is monotone. Next, on the one hand, since $T=P_{C}$, it follows from Theorem 3.2 that $f$ is Fréchet differentiable on $\mathcal{H}$ and $\nabla f={\operatorname{Id}}-T={\operatorname{Id}}-\sum_{i\in I}\alpha_{i}P_{C_{i}}$. On the other hand, for every $i\in I$, since $C_{i}$ is convex, closed, and nonempty, we infer from Theorem 3.2 (applied to $P_{C_{i}}$) that $d_{C_{i}}^{2}=2{\operatorname{q}}\circ({\operatorname{Id}}-P_{C_{i}})$ is Fréchet differentiable on $\mathcal{H}$ with $\nabla d_{C_{i}}^{2}=2({\operatorname{Id}}-P_{C_{i}})$. Altogether, since $\alpha=\sum_{i\in I}\alpha_{i}$ by definition, it follows from 43 that $g$ is Fréchet differentiable on $\mathcal{H}$ and that

and it thus follows that $f$ is Fréchet differentiable on $\mathcal{H}$ and, since $\alpha=\sum_{i\in I}\alpha_{i}$, $\nabla f=\sum_{i\in I}\alpha_{i}({\operatorname{Id}}-P_{C_{i}})-(\alpha-1){\operatorname{Id}}={\operatorname{Id}}-\sum_{i\in I}\alpha_{i}P_{C_{i}}={\operatorname{Id}}-T$. Hence, since $T$ is monotone by our assumption, Theorem 3.2 ensures the existence of a nonempty closed convex set $C$ such that $T=P_{C}$. Therefore, $f={\operatorname{q}}\circ({\operatorname{Id}}-P_{C})=(1/2)d_{C}^{2}$ and 41 follows from 45. ∎

As we have seen in Remark 2.5, the set $C$ in Theorem 3.10 need not be $\sum_{i\in I}\alpha_{i}C_{i}$.

We now establish a necessary and sufficient condition under which a finite sum of projectors is a projector.

in which case, $\sum_{i\in I}C_{i}$ is a closed convex set,

Since it is clear that $\sum_{i\in I}P_{C_{i}}$ is monotone, we derive from Theorem 3.10 (applied to $(C_{i})_{}$, $(\alpha_{i})_{}=(1)_{}$, and $\alpha=\operatorname{card}I=\sum_{i\in I}1$) and 9 that

According to Proposition 2.4(iii) and 50, we see that $\sum_{i\in I}C_{i}=C$ is a closed convex set, from which and 50 we get 47. Furthermore, it follows from 10 and 51 that

Let $C$ and $D$ be nonempty closed convex subsets of $\mathcal{H}$. Then the following are equivalent:

If (i) or (ii) holds, then $C+D$ is a closed convex set,

Consider the setting of Corollary 3.13. In view of [4, Example 12.3], we see that 54 is equivalent to (\iota_{C}\mbox{\footnotesize\,\square\,}\iota_{D})\mbox{\footnotesize\,\square\,}{\operatorname{q}}=\iota_{C}\mbox{\footnotesize\,\square\,}{\operatorname{q}}+\iota_{D}\mbox{\footnotesize\,\square\,}{\operatorname{q}}-{\operatorname{q}}+\gamma. Hence, using [4, Example 13.3(i) and Proposition 13.24(i)] and Moreau’s decomposition , we infer that

This type of relationship is used in [11, Proposition 3.16] to establish a condition for the sum of two proximity operators to be a proximity operator.

The following simple example shows that the constant $\gamma$ in Corollary 3.13 can take on any value.

Let $u$ and $v$ be in $\mathcal{H}$, set $C\coloneqq\{u\}$, and set $D\coloneqq\{v\}$. Then clearly $P_{C}+P_{D}=P_{\{u+v\}}=P_{C+D}$ and $(\forall x\in\mathcal{H})\,\langle{P_{C}x}\,|\,\mathopen{}{P_{D}x}\rangle=\langle{u}\,|\,\mathopen{}{v}\rangle$.

As a consequence of Corollary 3.13, a sum of projectors onto orthogonal sets is a projector; see [6, Proposition 2.6] for a difference derivation.

Let $C$ and $D$ be nonempty closed convex subsets of $\mathcal{H}$ such that $C\perp D$. Then the following hold:

$d_{C+D}^{2}=d_{C}^{2}+d_{D}^{2}-2{\operatorname{q}}$.

Since $(\forall x\in\mathcal{H})\,\langle{P_{C}x}\,|\,\mathopen{}{P_{D}x}\rangle=0$, the conclusions readily follow from Corollary 3.13. ∎

We now provide an instance where item (ii) of Corollary 3.13 holds, $C\nsubseteq D^{\perp}$ in general, and neither $C$ nor $D$ is a cone.

We next establish a necessary and sufficient condition for $u+P_{C}$ to be a projector.

Let $C$ be a nonempty closed convex subset of $\mathcal{H}$, and let $u\in\mathcal{H}.$ Then, since $(\forall x\in\mathcal{H})\,u=P_{\mathopen{}\left\{u\right\}}x$, we deduce from Corollary 3.13 that

in which case, $u+P_{C}=P_{u+C}$ due to Corollary 3.13.

Consider the setting of Example 3.18. Since $u+P_{C}$ is monotone, nonexpansive, and a sum of proximity operators, [2, Corollary 2.5] guarantees that $u+P_{C}$ is a proximity operator. However, by Example 3.18, it is not a projector unless $u\in(C-C)^{\perp}.$

Here is a sufficient, but not necessary, condition for a sum of projectors to be a projector.

Set $(\forall k\in I)\,D_{k}\coloneqq\sum_{i=1}^{k}C_{i}$, and let us establish that

Due to Corollary 3.13, the claim holds if $k=2$, and we therefore assume that, for some $k\in\{2,\ldots,m-1\}$, $D_{k}$ is a closed convex set and that $\sum_{i=1}^{k}P_{C_{i}}=P_{D_{k}}$. Then, by our assumption, $(\forall x\in\mathcal{H})\,\langle{P_{D_{k}}x}\,|\,\mathopen{}{P_{C_{k+1}}x}\rangle=\sum_{i=1}^{k}\langle{P_{C_{i}}x}\,|\,\mathopen{}{P_{C_{k+1}}x}\rangle=\sum_{i=1}^{k}\gamma_{i,k+1}$, from which and Corollary 3.13 (applied to $D_{k}$ and $C_{k+1}$) we infer that $D_{k+1}=D_{k}+C_{k+1}$ is a closed convex set and, due to the induction hypothesis, $\sum_{i=1}^{k+1}P_{C_{i}}=\sum_{i=1}^{k}P_{C_{i}}+P_{C_{k+1}}=P_{D_{k}}+P_{C_{k+1}}=P_{D_{k}+C_{k+1}}=P_{D_{k+1}}$. Hence, letting $k=m$ in 58 yields the conclusion. ∎

We now illustrate that the assumption of Corollary 3.20 need not hold when merely $\sum_{i\in I}P_{C_{i}}=P_{C}$.

Let $C$ be a nonempty closed convex subset of $\mathcal{H}$ such that $\mathcal{H}\smallsetminus(C-C)^{\perp}\neq\varnothing$, and suppose that $u\in\mathcal{H}\smallsetminus(C-C)^{\perp}.$ Then $P_{\mathopen{}\left\{u\right\}}+P_{\mathopen{}\left\{-u\right\}}+P_{C}=P_{C}$ is a projector. However, if $x\mapsto\langle{P_{\{u\}}x}\,|\,\mathopen{}{P_{C}x}\rangle=\langle{u}\,|\,\mathopen{}{P_{C}x}\rangle$ were a constant, then it would follow from Corollary 3.13 that $u+P_{C}=P_{\mathopen{}\left\{u\right\}}+P_{C}$ is a projector, which violates Example 3.18 and the assumption that $u\notin(C-C)^{\perp}.$

We conclude this section with a result concerning the difference of two projectors.

Convex combination of projectors

The analysis of this section requires the following results.

Let $(T_{i})_{}$ be a finite family of firmly nonexpansive operators from $\mathcal{H}$ to $\mathcal{H}$, let $(\alpha_{i})_{}$ be real numbers in $\mathopen{}\left]0,1\right]$ such that $\sum_{i\in I}\alpha_{i}=1$, and let $C$ be a nonempty closed convex subset of $\mathcal{H}$. Then $\sum_{i\in I}\alpha_{i}T_{i}=P_{C}$ if and only if there exist vectors $(u_{i})_{}$ in $\mathcal{H}$ such that $(\forall i\in I)\,T_{i}=P_{C}+u_{i}$ and $\sum_{i\in I}\alpha_{i}u_{i}=0$.

Let $C$ and $D$ be nonempty closed convex subsets of $\mathcal{H}$, and set $v\coloneqq P_{\overline{D-C}}0$. Then the following hold:

Let $(c_{n})_{}$ and $(d_{n})_{}$ be sequences in $C$ and $D$, respectively, and suppose that $d_{n}-c_{n}\to v$. Then $d_{n}-P_{C}d_{n}\to v$.

Suppose that there exists $u\in\mathcal{H}$ such that $P_{D}=P_{C}+u$. Then $u=v\in(C-C)^{\perp}$ and $D=C+v$.

(ii): Since $P_{D}=P_{C}+u$, Example 3.18 guarantees that $u\in(C-C)^{\perp}$ and that $D=C+u$. Hence, it suffices to show that $u=v$. Indeed, since $v=P_{\overline{D-C}}0\in\overline{D-C}$, there exist sequences $(c_{n})_{}$ in $C$ and $(d_{n})_{}$ in $D$ such that $d_{n}-c_{n}\to v$. Thus, we deduce from (i) that

Let $(C_{i})_{}$ be a finite family of nonempty closed convex subsets of $\mathcal{H}$, let $k\in I$, and set $(\forall i\in I)\,v_{i}\coloneqq P_{\overline{C_{i}-C_{k}}}0$. Then the following are equivalent:

For every $i\in I$, we have $v_{i}\in(C_{k}-C_{k})^{\perp}$ and $C_{i}=C_{k}+v_{i}$.

“(i)$\Rightarrow$(iii)”: Suppose that there exist $(\alpha_{i})_{}\in\mathopen{}\left]0,1\right]^{I}$ and a nonempty closed convex subset $C$ of $\mathcal{H}$ such that $\sum_{i\in I}\alpha_{i}=1$ and $\sum_{i\in I}\alpha_{i}P_{C_{i}}=P_{C}$. Then, since $(P_{C_{i}})_{}$ are firmly nonexpansive by [4, Proposition 4.16], Fact 4.1 guarantees the existence of vectors $(u_{i})_{}$ in $\mathcal{H}$ such that

Now fix $i\in I$. We then derive from 61 that $P_{C_{i}}=(P_{C_{k}}-u_{k})+u_{i}=P_{C_{k}}+u_{i}-u_{k}$, and it thus follows from Lemma 4.2(ii) (applied to $(C_{k},C_{i},u_{i}-u_{k})$) that $v_{i}\in(C_{k}-C_{k})^{\perp}$ and $C_{i}=C_{k}+v_{i}$, as required.

To complete the proof, we shall show that (ii)$\Leftrightarrow$(iii).

“(iii)$\Rightarrow$(ii)”: Suppose that (iii) holds. Then, due to 62, (iv) holds, from which (ii) follows. ∎

The following example shows that the conclusion of Theorem 4.3 fails if we replace “convex combination” by “affine combination” in item (i).

Let $C$ be a nonempty closed convex subset of $\mathcal{H}$, and let $u\in\mathcal{H}$. Then the affine combination of $(P_{C},P_{C},P_{\mathopen{}\left\{u\right\}})$ with weights $(1/4,-1/4,1)$ is a projector since $(1/4)P_{C}-(1/4)P_{C}+P_{\mathopen{}\left\{u\right\}}=P_{\mathopen{}\left\{u\right\}}$. However, Theorem 4.3(iii) fails when $C$ is not a singleton.

Here are some direct consequences of Theorem 4.3.

Let $k\in I$ and let $i\in I$. Since $C_{k}\cap C_{i}\neq\varnothing$ by assumption, we see that $P_{\overline{C_{i}-C_{k}}}0=0$, and thus, due to our assumption, the implication “(i)$\Rightarrow$(iii)” of Theorem 4.3 yields $C_{i}=C_{k}$, as desired. ∎

Let $C$ and $D$ be nonempty closed convex subsets of $\mathcal{H}$. Then the following are equivalent:

$P_{\overline{D-C}}0\in(C-C)^{\perp}$ and $D=C+P_{\overline{D-C}}0$.

This follows from the equivalences “(ii)$\Leftrightarrow$(iii)$\Leftrightarrow$(iv)” of Theorem 4.3. ∎

We now specialize Corollary 4.6 to get a result on scalar multiples of projectors.

Let $D=\mathopen{}\left\{0\right\}$ in Corollary 4.6. ∎

The partial sum property of projectors onto convex cones

In this section, we shall discuss the partial sum property and the connections between our work, Zarantonello’s [26, Theorems 5.5 and 5.3], and the recent work . We shall need the following two results. Let us provide an instance where the star-difference of two sets (see ) can be explicitly determined. Lemma 5.1 was mentioned in [2, Footnote 5] and was also stated implicitly in the proof of [26, Theorem 5.2].

Let $K_{1}$ and $K_{2}$ be nonempty closed convex cones in $\mathcal{H}$, and set

The chain of implications “(i)$\Rightarrow$(ii)$\Rightarrow$(iii)” is clear.

“(iv)$\Rightarrow$(i)”: First, take $u\in K_{1}$. Since $K_{2}\subseteq K_{1}$ and $K_{1}$ is a convex cone by assumption, it follows that $u+K_{2}\subseteq K_{1}+K_{1}\subseteq K_{1}$, and therefore $u\in K$. Conversely, fix $u\in K$. Because $u+K_{2}\subseteq K_{1}$ and $0\in K_{2}$, we deduce that $u\in K_{1}$, which completes the proof. ∎

Let $C$ and $D$ be nonempty closed convex subsets of $\mathcal{H}$, and set

(\forall u\in\mathcal{H})\,h(u)=\sup_{v\in D}\big{(}\sigma_{C}(u+v)+\langle{u}\,|\,\mathopen{}{v}\rangle\big{)}.

Suppose that $C$ and $D$ are cones and $D\subseteq{C}^{\ominus}.$ Then $h=\iota_{{C}^{\ominus}\cap{D}^{\ominus}}.$

(i): Since $D$ is convex, closed, and nonempty, we see that $\iota_{D}\in\varGamma_{0}(\mathcal{H})$, and so (1/2)d_{D}^{2}=\iota_{D}\mbox{\footnotesize\,\square\,}{\operatorname{q}}=\iota_{D}\mbox{\footnotesize\,\boxdot\,}{\operatorname{q}} by [4, Example 12.21 and Proposition 12.15]. In turn, Moreau’s decomposition asserts that {\operatorname{q}}-(1/2)d_{D}^{2}={\operatorname{q}}-\iota_{D}\mbox{\footnotesize\,\boxdot\,}{\operatorname{q}}=\iota_{D}^{\ast}\mbox{\footnotesize\,\boxdot\,}{\operatorname{q}}. Thus, 64 yields

Moreover, since $\iota_{D}\in\varGamma_{0}(\mathcal{H})$ and ${\operatorname{q}}^{\ast}={\operatorname{q}}$, [4, Proposition 13.24(i)] and the Fenchel–Moreau theorem guarantee that (\iota_{D}^{\ast}\mbox{\footnotesize\,\boxdot\,}{\operatorname{q}})^{\ast}=\iota_{D}^{\ast\ast}+{\operatorname{q}}^{\ast}=\iota_{D}+{\operatorname{q}}, which implies that \operatorname{dom}(\iota_{D}^{\ast}\mbox{\footnotesize\,\boxdot\,}{\operatorname{q}})^{\ast}=D. Consequently, because \iota_{D}^{\ast}\mbox{\footnotesize\,\boxdot\,}{\operatorname{q}}\in\varGamma_{0}(\mathcal{H}), [4, Proposition 14.19 and Example 13.27(iii)] imply that

(ii): First, because $D\subseteq{C}^{\ominus}$, Lemma 5.1 (applied to the pair of closed convex cones $({C}^{\ominus},D)$) yields

Next, we derive from (i) and [4, Example 13.3(ii)] that

Now fix $u\in\mathcal{H}$, and let us consider two alternatives.

(a) $u\in\mathcal{H}\smallsetminus{C}^{\ominus}$: In view of 66bo, there exists $v\in D$ such that $u+v\in\mathcal{H}\smallsetminus{C}^{\ominus}$, and therefore, by 66bp, $h(u)\geqslant\iota_{C^{\ominus}}(u+v)+\langle{u}\,|\,\mathopen{}{v}\rangle=+\infty.$

(b) $u\in{C}^{\ominus}$: Then, by 66bo, $u+D\subseteq{C}^{\ominus}$. Hence, since $D$ is a nonempty cone, it follows from 66bp and [4, Example 13.3(ii)] that $h(u)=\sup_{v\in D}\langle{u}\,|\,\mathopen{}{v}\rangle=\sigma_{D}(u)=\iota_{{D}^{\ominus}}(u)=\iota_{{C}^{\ominus}\cap{D}^{\ominus}}(u).$

Altogether, we obtain the desired conclusion. ∎

Here is the first main result of this section. The proof of the implication “(v)$\Rightarrow$(i)” was inspired by [2, Lemma 5.3].

Let $K_{1}$ and $K_{2}$ be nonempty closed convex cones in $\mathcal{H}$. Then the following are equivalent:

$K_{1}+K_{2}$ is closed and $P_{K_{1}}+P_{K_{2}}=P_{K_{1}+K_{2}}.$

There exists a nonempty closed convex cone $K$ such that $P_{K_{1}}+P_{K_{2}}=P_{K}$.

$P_{K_{1}}+P_{K_{2}}$ is a proximity operator of a function in $\varGamma_{0}(\mathcal{H})$.

${{\operatorname{Id}}}-P_{K_{1}}-P_{K_{2}}$ is monotone.