2.1. Norm Identity in and Its Consequences
In this subsection, we present our main results. Our first main result provides a new proof of the equality (
1), which relies primarily on two intrinsic properties of orthogonal projections: idempotence and positivity. To establish this proof, we require the following well-known lemmas.
The first lemma, stated in [
23] (Theorem 1.7 (b)), characterizes the norm of the sum of two operators whose ranges and adjoint ranges are mutually orthogonal.
Lemma 1. Let such that the ranges of S and T, and the ranges of and , are orthogonal to each other, respectively. Then The second lemma, see [
5,
6], asserts that the spectra of the products
and
coincide outside of the points 0 and 1, when
and
are orthogonal projections. This fact plays a crucial role in spectral comparisons arising in our analysis.
Lemma 2 ([
5] (Lemma 3) or [
6] (Equation (2.5))
. Let . Then We are now in a position to present our first result in this work.
Theorem 1. Let such that neither nor coincides with . Then it follows that Proof. Let us begin by noting that the self-adjoint operator
can be decomposed as follows:
where
and
.
Then, from the fact that
Q is an idempotent operator, it follows that
and
From the previous equalities, we can conclude that the ranges of
S and
T, as well as the ranges of
and
, are orthogonal to each other, respectively. Then by Lemma 1, we conclude that
Since
, it follows immediately that
, with equality holding only in the special case that
. We now examine the case where
and
To prove the equality between the norms of Q and , we proceed by contradiction.
Assume, without loss of generality, that
Since
, both
Q and
are bounded linear operators, and their products with their adjoints are positive and self-adjoint. In particular, the operators
are positive and normal, so their norms coincide with their spectral radii:
Therefore, from our assumption,
Now, consider the spectral properties of the operators
and
. By Lemma 2, applied to the pair
,
, we know that the nonzero spectral values of
and
, other than possibly 0 and 1, coincide:
But since the spectrum of a positive operator lies in
, the inequality
would imply that a spectral value of
, outside of
, exceeds all the spectral values of
, contradicting the equality of spectra stated above.
Therefore, our assumption must be false. We conclude that
The case
is entirely analogous; it leads to the reversed inequality
, and the same contradiction with the spectral correspondence given by Lemma 2 arises.
In conclusion, we have proven that
This completes the proof. □
Remark 1. We clarify why we must assume that neither nor coincides with for the equalityto hold. If , then , and hence while . On the other hand, if , then, since Q is idempotent, we have that for every . Indeed, if , there exists such that , and thusIn this case, and . Before proceeding, let us illustrate with an example that the norm equality (
1) is not a property exclusive to bounded idempotent operators.
Remark 2. Let be the Hilbert space of square-summable sequences. We aim to construct a bounded linear operator P defined on such that and satisfies .
Define the operator P as follows:This operator is bounded and linear. However, it is not idempotent, since . For example, consider the sequence : Furthermore, it is trivial to see that . Now, we will demonstrate that the range of P does not coincide with . Consider the sequence with . Clearly, . However, if , then for all , which implies thatThus, , and does not belong to the range of P. Finally, it is immediate to verify that Corollary 1. Let such that neither nor coincides with . Then it follows that Proof. Let us first verify that if , then as well, and that it satisfies the hypotheses of Theorem 1.
Suppose, for contradiction, that
. Then
for all
. By the definition of the adjoint, we have
which implies that
for all
, and hence
, a contradiction.
On the other hand, if , then, as noted in Remark 1, it follows that , and consequently , which contradicts the assumption that .
We thus conclude that
, and in particular, neither
nor
coincides with
. Therefore, by Theorem 1, we obtain
the desired conclusion follows. □
As a consequence of Lemma 1, we have the following norm equalitites.
Corollary 2. Let such that neither nor coincides with . Then it follows that Proof. Let us note that it is sufficient to prove the last equality. Let us consider
and
, then
where the ranges of
S and
T, as well as the ranges of
and
, are orthogonal to each other, respectively. Thus, applying Lemma 1, we deduce that
Hence, the proof is complete. □
Combining Theorem 1 and Corollary 2, we can assert that if
is such that neither the null space
nor the range
coincides with the whole space
, then the following identities hold:
where
denotes the real part of the operator
.
Analogously, considering the imaginary part
, we also obtain:
Moreover, we may conclude that
i.e., for any
such that neither
nor
coincides with
, it holds that
This means that the norm of the sum of the two operators
and
is equal to the sum of their norms. In [
24], the authors provide a characterization of when such an equality holds for arbitrary pairs of operators in
, and we therefore recommend their work for further reading.
Let
, and let
and
denote the orthogonal projections onto these subspaces. In [
7], Buckholtz established necessary and sufficient conditions to guarantee the existence of an operator
with
and
, respectively. Furthermore, in the same article, Buckholtz derived an explicit formula for
, with
, in terms of the orthogonal projections
and
. More precisely,
It is worth mentioning that the proof presented by Buckholtz of the identity (
3) was not the first. For this reason, in [
17], Szyld provides a historical overview of the various proofs for such equality, particularly in Sections 6 and 7.
We now present an identity for the norm of Q in terms of the norm of the sum and the anticommutator of the orthogonal projections and .
Let us recall that if
, the anticommutator of
S and
T is defined as
. Recently, Walters, in [
13], derived a formula for the norm of the anticommutator in the context of orthogonal projections. In particular, if
, then the norm of the anticommutator
is expressed as a simple quadratic function of
, more precisely
Theorem 2. Let with range and kernel . Then, if , we have Proof. Since, by assumption, we are excluding the trivial case
, we may assert that the orthogonal projection onto the range of
Q, namely
, is nonzero. In this setting, we invoke the Duncan–Taylor inequality, which establishes a sharp norm identity for the sum of two orthogonal projections (see [
22]). Specifically, applying this result to the pair
and
, we obtain
Now, by combining this expression with the identity given in (
4), we can derive the following chain of equalities:
In conclusion, this final identity, in conjunction with the previously established norm formula in (
3), allows us to complete the proof directly and without further elaboration. □
Using the recently obtained identity for the norm of Q, we can derive the following characterization for Q to be an orthogonal projection.
Corollary 3. Let , with . Then, the following statements are equivalent:
1. , i.e., .
2.
3. .
Proof. Suppose that
, with
. In particular, this ensures that either
or
. By (
3), we have
which leads to
. Consequently, using Duncan-Taylor’s identity for the norm of the sum of orthogonal projections, we deduce that
. Furthermore, by [
21] (Proposition 2.9), it follows that
Observe that if , then by Duncan-Taylor’s formula, .
Finally, when , we conclude that with , i.e., . □
In [
4], the renowned Krein–Krasnoselskii–Milman equality (KKME) was derived. It establishes a precise relationship for the norm of the difference between two orthogonal projections in terms of their interactions with complementary subspaces. More precisely, if
, then it holds
We now present a generalization of KKME within the framework of oblique projections. Unlike the original KKME, which is restricted to orthogonal projections, this extended result captures analogous properties for oblique projections, thereby broadening its applicability to a wider class of operators.
Theorem 3. Let . Then, we have Proof. Define
and
. Notice that
and
Thus, the ranges of
T and
S, as well as those of
and
, are pairwise orthogonal. By Lemma 1, we conclude that
So, the proof is finished. □
Corollary 4. Let such that . Then, the following equality holds: Before proceeding, we will first demonstrate, through a simple example, that the result presented above is indeed a generalization of KKME. More specifically, we will prove that there exist operators that satisfy the condition
Let
It is trivial to verify that
and
. Furthermore, since neither
nor
is self-adjoint, we conclude that
.
In conclusion, the matrices and are oblique projections (but not orthogonal projections), and they satisfy the condition:
In particular, if , the hypotheses of Corollary 4 are immediately satisfied, and thus, the KKME follows as a consequence.
Proposition 1. Sean , then it holds 2.2. The Operator Distances from Oblique Projections to Orthogonal Projections
We now turn our attention we turn our attention to studying the distance from a given oblique projection
Q to the set
of all orthogonal projections. We recall that Zhang, Tian, and Xu recently provided a complete characterization of the operator distances from an idempotent operator to the set of orthogonal projections, including the minimum, maximum, and intermediate values (see [
19]). Before recalling this result, we must review the notion of generalized inverse for bounded linear operators.
Let
. An operator
S is said to be a generalized inverse of
T if it satisfies the conditions
Such operators arise naturally in the study of linear equations, approximation theory, and optimization, especially in the context of solving inconsistent or underdetermined systems. It is important to emphasize that the generalized inverse of a given operator is not unique in general.
A particularlysignificant instance of a generalized inverse is the Moore–Penrose inverse. The operator
S is called the Moore–Penrose inverse of
T if, in addition to the two identities above, it also satisfies the symmetry conditions
When it exists, the Moore–Penrose inverse is unique and is denoted by
. It plays a central role in functional analysis and operator theory, as it generalizes the notion of the inverse to non-invertible or non-square operators in a canonical way.
It is well known (see [
25]) that an operator
T admits a Moore–Penrose inverse if and only if its range is closed. Moreover, if
T is invertible, then the Moore–Penrose inverse coincides with the usual inverse, i.e.,
. For a comprehensive treatment of generalized inverses in Hilbert spaces, we refer the interested reader to [
26].
More precisely, they obtained the following results.
Lemma 3 ([
19] Theorem 2.3 and 2.4)
. Let . The distance is the minimum distance from Q to all orthogonal projections on , whereand . If is such that is the minimum distance from Q to all projections on , thenwhich represents the maximum distance from Q to all projections on . Remark 3. If , then from the identities and the fact that Q commutes with , together with the equalityit follows almost immediately thatIn the more general case where , the explicit computation of the orthogonal projection closest to the oblique projection Q requires deeper insight. For a detailed treatment of this problem, we refer the reader to the recent manuscript [19]. In particular, the motivation behind this approach is clearly articulated in Problem 1.1 of that work, where the authors begin by analyzing the problem explicitly in the setting of , i.e., complex matrices. In that specific context, the minimizing orthogonal projection is computed in closed form. The general formula for , valid for any Hilbert space , is then obtained by extending the techniques developed for matrices and applying operator-theoretic tools. In particular, the authors rely heavily on the block matrix representation of an operator relative to a projection, which plays a central role in deriving the general result.
As an immediate consequence of Theorem 1 and Lemma 3, we can conclude the following result.
Proposition 2. Let be such that neither nor coincides with . Let be an orthogonal projection minimizing the distance to Q, i.e.,Then, and . Proof. Suppose that there exists
Q with the given hypotheses, and by reductio ad absurdum, assume that
or
. Then, by (
6), it follows that
if
, or
if
. Notice that both equalities contradict the fact that the norms of
Q and
coincide. Therefore, we can conclude that
and
. □
From the previous result, we derive the following inequality. It is worth noting that the same inequality was proved in [
18] (Corollary 3.13), using a different approach. Furthermore, our inequality is strict for every
whose null space and range are both distinct from the entire Hilbert space.
Corollary 5. Let such that neither nor coincides with , then Taking into account that, in [
18], the authors recently derived an explicit formula for the norm of
, we can establish the following refinement of (
7).
Theorem 4. Let such that neither nor coincides with , then Proof. By [
18] (Theorem 3.17), we have that
We first rewrite the expression:
Now, applying the inequality
which follows from the concavity of the square root function, we obtain
This finishes the proof. □
Remark 4. We will show that the first inequality established in Theorem 4 is strict whenever .
Suppose, by contradiction, that there exists , with neither nor equal to , such that By identity (
8)
, this is equivalent to Squaring both sides and performing standard algebraic manipulations, we obtain Since we assumed that , we may cancel the common factor and deduce that Squaring once more, we arrive at , a contradiction.
Therefore, we have the following statement
Corollary 6. Let , with neither nor equal to , and , then At this stage, we are in a position to establish a lower bound for the minimum distance between an idempotent operator and the set of orthogonal projections .
Proposition 3. Let . ThenFurthermore, equality holds if and only if . Proof. If , the inequality is trivial.
Hence, we may assume that .
Then, by identity (
8) and the AM–GM inequality, we obtain
Then, the identity (
8) immediately yields
so that
.
Conversely, if
, then clearly
, and hence
This finishes the proof. □
Remark 5. In Proposition 3, we proved that if , then Suppose, for the sake of contradiction, that there exists an operator satisfying (
10)
, that is,with . Then,and since , we conclude that Equivalently,which implies thata contradiction. Combining Proposition 3 with the preceding observation, we arrive at the following result.
Given that if
with
, then
, it follows that
By employing the matched operator and the inequalities established earlier, we can derive the following refinement, which we present in the proposition below.
Proposition 4. Let be such that neither nor coincides with . Then,In particular,and if , the inequality is strict. We conclude this section by deriving upper bounds for the norm of
, where
. If
, it is straightforward to verify that
. Observe that, based on (
8), we obtain
for any
.
This bound is not sharp since
Indeed, suppose, for the sake of contradiction, that
or equivalently,
However, this inequality implies that
, which contradicts (
9).
Next, we present an upper bound for the norm of , which is sharper than the one given by the triangle inequality.
Theorem 5. Let such that neither nor coincide with . Then,Furthermore, equality holds if and only if . Proof. As a consequence of the positivity of
and
I, and using the equality (
2), we conclude that
and
, respectively.
Thus,
We conclude the proof by observing that if we consider the functions
where
. Then,
for any
.
Now suppose that equality holds, namely,
Then, by (
11), we have
which implies that
, and hence
. Conversely, if
, then Duncan–Taylor’s formula implies
This finishes the proof. □
It is well known that if then P is self–adjoint and hence normal. Every normal operator satisfies that is, P is normaloid. By contrast, we will show that oblique non orthogonal projections do not necessarily share this property.
Corollary 8. Let . Then . In particular, Q is not normaloid.
Proof. Suppose, to the contrary, that
Q is normaloid, i.e.,
. By the definition of the numerical radius, there exists a sequence of unit vectors
such that
For each
n, choose
so that
Since the unit circle
is compact, by Bolzano–Weierstrass there is a subsequence
converging to some
. Hence
It follows that
, and in particular,
. By [
24] (Theorem 2.1), this implies
But then Theorem 5 forces
, contradicting
. Therefore, no oblique projection
can be normaloid. □
Remark 6. We observe that an alternative proof of Corollary 8 follows from the classical characterization of normaloid operators (see [2]): an operator is normaloid if and only if . Indeed, if were normaloid, then for every one would havesince by idempotence. It follows that , and hence , a contradiction. While this concise argument is elegant, we have chosen to develop and present proofs based on the results established earlier in this manuscript, as they offer deeper insight into the geometric and spectral properties of oblique projections. In Theorem 5, we show that if
, then
Remark 7. Within the framework of bounded linear operators on Hilbert spaces, the identitywith , is referred to as the Daugavet equation; see, for example, [27,28] and the references therein. In particular, we have shown that no operator satisfies the Daugavet equation. We now show that non-orthogonal oblique projections satisfy an inequality that extends (
12). More precisely:
Theorem 6. Let . Thenfor any Proof. Suppose that there exists
such that
Then, by [
29] (Proposition 4.7), we conclude that
Q is normaloid, which contradicts Corollary 8. This completes the proof. □