We show that the two main results of the article [J. Math. Phys. 47, 102103 (2006)] have very short proofs as direct consequences of the solution to the infimum problem for bounded non-negative operators in a Hilbert space given by T. Ando [Analytic and Geometric Inequalities and Applications, Mathematical Applications Vol. 478 (Kluwer Academic, Dordrecht, 1999)] and a formula for the shorted operator obtained by H. Kosaki [“Remarks on Lebesgue-type decomposition of positive operators,” J. Oper. Theory 11, 137–143 (1984)].

Let $H$ be a (complex) Hilbert space and $B(H)$ the $C*$-algebra of bounded linear operators in $H$. The cone of bounded non-negative operators in $H$ is denoted by $B(H)+$. The order relation ⩽ is the natural one: Given $A,B\u220aB(H)+$, $A\u2a7dB$ if and only if $\u27e8Ah,h\u27e9\u2a7d\u27e8Bh,h\u27e9$ for all $h\u220aH$.

Given $A,B\u220aB(H)+$, the *parallel sum* of $A$ and $B$ is the bounded non-negative operator denoted by $A:B$ and defined (e.g., see Ref. 6) by

Parallel addition is separately nondecreasing and $0\u2a7dA:B\u2a7dA,B$.

The *shorted operator* of $A$ by $B$ is the bounded non-negative operator denoted by $[A]B$ and defined (e.g., see, Ref. 1) by

where the strong operator limit exists because $(nA):B\u2a7dB$ and $(nA):B\u2a7d((n+1)A):B$ for all $n\u220aN$. Note that $[A]B\u2a7dB$ but $[A]B$ may not be comparable with $A$.

Another formula to calculate $[A]B$ is available. Namely, let $PA,B$ denote the orthogonal projection of $H$ onto the closure of the subspace ${h\u220aH\u2223B1\u22152h\u220aRan(A1\u22152)}$, where $RanC$ denotes the range of the operator $C$.

**Theorem 1:** (Reference 5). *If* $A,B\u220aB(H)+$ *then* $[A]B=B1\u22152PA,BB1\u22152$.

Given $A,B\u220aB(H)+$, let $A\u2227B$ denote the *infimum* of $A$ and $B$, if it exists, and defined as the greatest lower bound of the set ${A,B}$ in the partially ordered set $B(H)+$, that is, $A\u2227B\u2a7dA,B$ and, if $C\u220aB(H)+$ is such that $C\u2a7dA,B$, then $C\u2a7dA\u2227B$. Questions as characterizations of the existence and computation of the infimum operator are related to the lattice properties of quantum effects, see Refs. 3 and 4. The most comprehensive answer is the following.

**Theorem 2:** (Reference 2). *Let* $A,B\u220aB(H)+$. *Then* $A\u2227B$ *exists if and only if the shorted operators* $[A]B$ *and* $[B]A$ *are comparable, that is, either* $[A]B\u2a7d[B]A$ *or* $[B]A\u2a7d[A]B$. *In this case*, $A\u2227B=min{[A]B,[B]A}$.

In this short note we show that the two main results of Ref. 7 can be easily derived from Theorems 2 and 1. The first one is the positive answer to a conjecture in Ref. 3.

**Theorem 3:** (Reference 7). *Let* $A,B\u220aB(H)+$ *be injective and at least one of them invertible. Then* $A\u2227B$ *exists if and only if* $A$ *and* $B$ *are comparable*.

*Proof*: Assume that $A$ is injective and $B$ is invertible. Then $B1\u22152$ is invertible and $Ran(B1\u22152)=H$. Thus, ${h\u220aH\u2223A1\u22152h\u220aRan(B1\u22152)}=H$ and hence $PB,A=I$. In particular, by Theorem 1 we have $[B]A=A$.

On the other hand,

and then taking into account that $A1\u22152$ is injective and hence, that it has dense range, it follows that $PA,B=I$. Therefore, $[A]B=B$. The statement follows now from Theorem 2.◼

For $C\u220aB(H)$ we denote by $\sigma (C)$ the spectrum of the operator $A$. The latter main result of Ref. 7 is the following.

**Theorem 4**: (Reference 7). *Let* $B\u220aB(H)+$. *Then* $I\u2227B$ *exists if and only if either* $\sigma (B)\u2286{0}\u222a[1,\Vert B\Vert ]$ *or* $\sigma (B)\u2286[0,1]$.

*Proof:* Because $Ker(B)$ is reducing both the identity operator $I$ and $B$, without loss of generality we can assume that $B$ is injective. In this case, the statement to be proven becomes as follows: *if* $B\u220aB(H)+$ *is injective then* $I\u2227B$ *exists if and only if either* $\sigma (B)\u2286[0,1]$ *or* $\sigma (B)\u2286[1,\Vert B\Vert ]$. Elementary spectral theory shows that $\sigma (B)\u2286[0,1]$ is equivalent with $B\u2a7dI$, while $\sigma (B)\u2286[1,\Vert B\Vert ]$ is equivalent with $I\u2a7dB$. Thus, the statement to be proven is a particular case of Theorem 3.◼