Abstract
Using the Dixmier angle between two closed subspaces of a complex Hilbert space , we establish the necessary and sufficient conditions for the operator norm of the sum of two orthogonal projections, and , onto closed subspaces and , to attain its maximum, namely These conditions are expressed in terms of the geometric relationship and symmetry between the ranges of the projections. We apply these results to orthogonal projections associated with a closed-range operator via its Moore–Penrose inverse. Additionally, for any bounded operator T with closed range in , we derive sufficient conditions ensuring where denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges and their algebraic structure governs norm extremality and extends a recent finite-dimensional result to the general Hilbert space setting.
Keywords:
orthogonal projections; symmetry; norm of projection sums; Dixmier angle; Moore–Penrose inverse; closed-range operators; Hilbert spaces MSC:
47A05; 47A30; 46C05; 47B15; 15A60
1. Introduction
Orthogonal projections are fundamental in functional analysis and operator theory, appearing in a wide range of theoretical and applied contexts. As self-adjoint idempotent operators, they are symmetric in the operator-theoretic sense, since each projection coincides with its adjoint. This intrinsic symmetry plays a crucial role in understanding their behavior, especially when analyzing sums of projections and their norm properties. Foundational results on the norm of projection sums are presented in [,], while various norm inequalities involving projections and related operators are explored in [,,], often revealing deeper geometric and algebraic symmetries. Further structural insights into projection operators and their ranges are given in [,]. In addition, the Moore–Penrose inverse is a key tool in the study of closed-range operators, with its properties—particularly in forming symmetric operator expressions like —examined in [,,].
This work studies the operator norm of the sum of two orthogonal projections, , where and denote the projections onto closed subspaces and of a complex Hilbert space . Our goal is to find conditions under which this norm reaches its maximum value, , and to apply these results to projections associated with bounded operators and their Moore–Penrose inverses. This extends recent finite-dimensional results [,] to infinite-dimensional Hilbert spaces, addressing a significant open problem in operator theory.
A crucial result by Duncan and Taylor [] shows that
This identity has been studied in finite-dimensional spaces [,] and within the framework of two-projection theory [], but the infinite-dimensional case remains challenging. The Dixmier angle, explored by Deutsch [], offers a geometric tool for analyzing subspace relationships and guides our characterizations. Works on operator ranges [] and range inclusion [] also support our study.
A key motivation is to connect these ideas to bounded operators with closed ranges and their Moore–Penrose inverses, which are fundamental in operator theory [,]. The Moore–Penrose inverse, detailed in [,], plays a central role in studying projections onto ranges and kernels of operators. Inspired by Conde’s recent work [] on projection norm maximization, we explore conditions under which
where T is a bounded operator with closed range and denotes its Moore–Penrose inverse. This generalizes finite-dimensional results [] to infinite-dimensional spaces, despite challenges such as non-compact operators and unbounded inverses [,]. We suggest that the reader consult the following manuscripts for further details and background on the topic [,,].
Next, we introduce the key concepts and notations used in this paper.
Let be a complex Hilbert space. We denote by the -algebra of all bounded linear operators on , equipped with the operator norm. For , we write for its adjoint, for its kernel, and for its range.
An operator is positive if for all . This implies T is self-adjoint, i.e., . The set of positive operators is denoted by .
An operator is an orthogonal projection if and . Such projections are essential for studying subspaces of and their geometry.
The subset consists of operators in with closed range. An operator T is normal if . Every orthogonal projection is normal and has closed range.
The spectral radius of is
where is the spectrum of T. The numerical radius is
The Dixmier angle, introduced in [], measures the relative position of two closed subspaces . It is defined as
This quantity is crucial in operator theory and spectral analysis.
We work with various classes of operators in . Beyond normality, several classes of non-normal operators have distinct structural and spectral properties. An operator is defined as follows:
- Normaloid if .
- Spectraloid if .
These classes are related by the following inclusions:
For an operator , the Moore–Penrose inverse is the unique operator satisfying the Penrose equations:
For more details, see [].
Define the operators
These are orthogonal projections onto and , respectively. Then, we have
We conclude by recalling a notable subclass within the set of closed-range operators. An operator is called an EP operator if it has closed range and commutes with its Moore–Penrose inverse, that is, , where denotes the usual commutator in ; see [].
2. Main Results
In this section, we provide necessary and sufficient conditions for the norm of the sum of two orthogonal projections to attain its maximum value. This characterization will be formulated in terms of the geometry of their ranges, or, equivalently, in terms of the closedness of certain sums or products of the associated projections.
To obtain our main result, we rely on two key corollaries. The first derives from a general identity concerning the sum of operator ranges—originally due to Crimmins (unpublished) and later simplified by Fillmore and Williams []. The second extends this result specifically to the setting of orthogonal projections.
Lemma 1
(Corollary 3 in []). Let . Then if and only if is closed.
Lemma 2
(Corollary X in []). be orthogonal projections on . Then, the following conditions are equivalent:
- 1.
- is closed.
- 2.
- .
- 3.
- .
- 4.
- .
Using these results, we are ready to prove one of the main theorems of this section.
Theorem 1.
Let be orthogonal projections on . Then, the following conditions are equivalent.
- 1.
- .
- 2.
- or is not closed.
- 3.
- or .
- 4.
- or .
- 5.
- or .
- 6.
- or .
Proof.
The quantity corresponds to the cosine of the Dixmier angle between the subspaces and , as defined in []:
From [], Lemma 10, we know that
if and only if either or is not closed. This establishes the equivalence between statements (1) and (2).
Statements (3) to (6) all relate the closure of sums and differences of projections and the closedness of ranges of corresponding operators.
By Lemma 1, we know that for positive operators , the sum has closed range if and only if the sum of their ranges is closed. Here, projections and satisfy these conditions.
Furthermore, Lemma 2 states that for orthogonal projections and , the condition that the sum of subspaces is closed is equivalent to the operators and having closed range, as well as to the operators and having closed range.
Thus, the closedness of is equivalent to each of the conditions that the operators in (3), (4), (5), and (6) belong to .
Therefore, the negation of the closedness condition in (2), i.e., is not closed, is equivalent to each of the operators listed in (3) to (6) not having closed range. The intersection condition is equivalent to
so all statements consistently use the ranges of projections. In conclusion, this proves the equivalence between statements (2) and (3), (4), (5), and (6). □
Remark 1.
Before presenting a new characterization, we recall a result previously obtained by one of the authors (see []).
Lemma 3.
Let be the orthogonal projections on . Then, the following conditions are equivalent.
- 1.
- 2.
- 3.
- 4.
- 5.
- for any
- 6.
Note that the condition that the sum of the ranges is a not closed set, as stated in Theorem 1, is superfluous when these subspaces are finite-dimensional. More precisely, we have the following result:
Corollary 1.
Let and be orthogonal projections on , where and are finite-dimensional subspaces. Then, the equality holds if and only if
Proof.
The result follows immediately from the well-known fact that if and are finite-dimensional subspaces, then their sum is closed, together with Theorem 1. □
We will now demonstrate that the hypotheses given in the obtained characterization in Theorem 1 are independent in an infinite-dimensional setting. Specifically, we seek three examples that illustrate this independence: (1) but is closed, (2) but is not closed, and (3) both and are not closed.
Moreover, in each example, we compute the norm of , where and denote the orthogonal projections onto and , respectively.
Example 1.
1. Let be the space of square-summable sequences, and let be its canonical basis. Consider the following closed subspaces of :
- It is immediately clear that , meaning that the sum of and is closed. Let and denote the respective orthogonal projections onto and . In this case, the intersection of their ranges is given by
- which is a proper subspace of . Therefore, .
- Moreover, applying to yields Thus, we conclude that .
- 2.
- Let be a Hilbert space with orthonormal basis . For each , defineConsider the closed subspaces andIt is evident that every basis vector belongs to , so the closure of is the entire space . However, one can show that the vector does not belong to , and therefore . To see this, assume that we can expresswhere and A careful examination shows that we must have and consequently for every . However, this is a contradiction since both and must belong to , implying thatand hence the sum is not closed.Moreover, it is straightforward to verify that . Indeed, if , then u can be written both as Since the decomposition is unique, we deduce that for all n. This forcesand hence all must vanish, so that .Finally, note that . This fact, along with standard arguments on the interplay between projections, shows that or equivalently .
- 3.
- Let be a Hilbert space with an orthonormal basis . Consider the closed subspacesUsing a similar argument to the one in the previous example, we can conclude that is not closed. Moreover, it is trivial that , which implies that the intersection of these subspaces is nontrivial. In particular, we conclude that
Remark 2.
It is worth mentioning that Example 1 (2) was previously presented in [] as a counterexample to a result discussed in that paper (see Proposition 2.6).
We are now in a position to characterize precisely when the norm of the sum of two orthogonal projections attains its maximal value.
Theorem 2.
Let be orthogonal projections on . Then, the following assertions are equivalent:
- 1.
- ;
- 2.
- There exists a sequence of unit vectors such that
Proof.
Assume that . By the definition of the operator norm, there exists a sequence of unit vectors such that
Since the sequences and are bounded in , we may extract subsequences—still denoted by —such that both norms converge. Now observe the following:
Since the first and last expressions are equal, it follows that the inequalities in between must actually hold as equalities. Therefore, we conclude that
and
Now for the converse direction. Suppose there exists a sequence , with for all n, such that
For each n, we have
Taking limits as gives
Therefore
Since it is well known that and we have exhibited unit vectors for which
it follows that
as claimed. □
As a consequence of Theorem 2, we obtain the following result, which provides new characterizations for when the norm of the sum of two orthogonal projections attains its maximum value.
Corollary 2.
Let be orthogonal projections on . Then, the following conditions are equivalent.
- 1.
- 2.
- There exists a sequence of unit vectors such thatwhere denotes the real part of .
- 3.
- 4.
Proof.
Assume that (1) holds. By Theorem 2, there exists a sequence of unit vectors such that
and
Therefore,
which implies that
If (2) holds, and noting that the operator is self-adjoint, we obtain
and hence,
Suppose that the norm of the anticommutator is equal to 2, then we have
which implies
Finally, if (4) holds, and since , it follows that . Then, by (1), we conclude that
□
In order to obtain a new characterization of when the sum of two orthogonal projections is maximal in norm, we begin by recalling the following lemma. We use the notion of Birkhoff–James orthogonality: for elements in a normed linear space , we say that u is orthogonal to v in the sense of Birkhoff–James, denoted , if
Lemma 4
(Proposition 4.1 in []). Let be a normed linear space. For , the following statements are equivalent:
- 1.
- .
- 2.
- .
- 3.
- .
Theorem 3.
Let be orthogonal projections on . Then if and only if there exists a sequence of unit vectors such that
Proof.
Assume that . Then, by the triangle inequality, we have
which implies in particular that
Using Lemma 4, we conclude that
Since is a positive operator, by the characterization of orthogonality in given in [], Lemma 1, it follows that there exists a sequence of unit vectors such that
Note that, since orthogonal projections are idempotent, we have
and therefore we can conclude that the sequence satisfies
Conversely, if there exists a sequence of unit vectors such that (3) holds, then applying again [], Lemma 1 together with Lemma 4, we conclude that
□
Remark 3.
taking limits gives
hence Moreover,
so by passing to the limit,
and therefore
It is clear that the conditions obtained in Theorems 2 and 3 are equivalent to each other via the equality . However, for the reader’s better understanding, we present the equivalence explicitly below.
- Let us consider the following statements:
- 1.
- There exists a sequence of unit vectors , , such that
- 2.
- There exists a sequence of unit vectors such that
Conversely, if , then we have
so . And since both and , it follows that
For the sequel, we aim to investigate the case when T is a closed-range operator.
Let with . Since and are orthogonal projections, it follows from (1) that
In this section, we explore conditions on the operator T to ensure that
A direct characterization of this equality, derived from (4) (or Proposition 3), is as follows:
indicating that the product of the orthogonal projections and has norm 1.
An intuitive assumption to explore is the containment relationship between the ranges of the projections, specifically
Since these subspaces are closed, we express this condition in terms of their kernels.
Theorem 4.
Let with satisfy
Then,
Proof.
Assume and suppose, for contradiction, that Since T has closed range, it follows that
Thus, we obtain which contradicts the assumption that . Therefore, Applying Theorem 1, we conclude that □
The inclusion conditions on the kernels stated in the previous theorem can be expressed as a well-known consequence of Douglas’ theorem on range inclusions (see []). Specifically, given , the following equivalences hold:
or
For operators with closed range that satisfy the equality , the following result follows directly from the preceding theorem.
Corollary 3.
Let with such that Then, .
Remark 4.
We can provide an ad hoc proof for closed-range operators T whose kernel coincides with that of their adjoint. More precisely, if T satisfies these conditions, Brock showed in [] that the equality is equivalent to the commutativity of T and its Moore–Penrose inverse, namely, Consequently,
since is an orthogonal projection.
The closed-range operators that satisfy the commutativity condition mentioned above have been extensively studied and are referred to as EP operators (see []). More precisely, given , we say that T is an EP operator if and only if its range coincides with that of its adjoint, i.e.,
To conclude this remark, we mention that in [], a characterization of the equality of the kernels of T and , where T is a closed-range operator, was obtained. This result is closely related to the previously discussed theorem of Douglas concerning the inclusion of ranges. Specifically, it is shown that
In conclusion, we give the following statement for EP operators.
Proposition 1.
Let T be an EP operator with . Then,
To support the preceding discussion, we now present an example of an infinite-dimensional operator in the EP class and verify the assertions stated above.
Example 2.
Let , and consider the operator defined by
This operator can be represented by the infinite matrix
In this case, the operator T has range , which is closed, implying that T belongs to the class . Moreover, T is a self-adjoint and idempotent operator, i.e., an orthogonal projection. As a consequence, its Moore–Penrose inverse coincides with itself, , and thus it commutes with its Moore–Penrose inverse, satisfying . Therefore, T is an EP operator. However, T is not invertible since is infinite-dimensional.
Let us now compute the norm . Since , we have the following:
and thus,
We obtain the following corollary for normal operators.
Corollary 4.
Let be a normal operator with . Then, we obtain .
Proof.
Since T is normal, it satisfies , which implies that . Consequently, by Corollary 3, we conclude that □
Recently, in [], new bounds for the numerical radius were obtained using the Moore–Penrose inverse, refining several previously known estimates. More precisely, if , then we have
This inequality represents a refinement, within the class of closed-range operators, of the classical bound established by Kittaneh in [],
which holds for every .
In particular, in [, Theorem 2.4], the authors established necessary and sufficient conditions for the validity of (5) in the finite-dimensional setting.
Lemma 5
(Proposition 2.1 in []). Let , with finite-dimensional and . Then, if and only .
It is worth mentioning that in [], the authors provide two different proofs of the preceding result. The first proof uses the fact that , while the second is based on the observation that both P and Q have 1 as an eigenvalue and share a common eigenvector. However, the result can also be deduced from Corollary 1 by considering and .
In the following result, we establish that if a closed-range operator T attains Kittaneh’s upper bound, then it also satisfies (5).
Theorem 5.
Let , with being such that
Then, or is not closed.
Proof.
Suppose, for the sake of contradiction, that and is closed. Equivalently, by Theorem 1, we deduce that Then, by inequality (6), it follows that
This is a contradiction, and hence we conclude that or is not closed. □
As an immediate consequence of the previous result, the following holds.
Corollary 5.
Let , , and T normaloid. Then,
Proof.
The fact that T is normaloid implies that . Then,
which means that Thus, by Theorem 5, we conclude that or is not closed. By Theorem 1, we conclude that □
In the following statement, we see that with the additional condition that the Hilbert space is finite-dimensional, the previous Corollary is more precise regarding its conclusion.
Proposition 2.
Let , where is finite-dimensional, with T being normaloid and . Then, we conclude that and, in particular,
Proof.
Since is finite-dimensional, it follows that , and therefore, it satisfies the hypotheses of Corollary 5. Consequently, we can assert that either or is not closed. However, due to finite dimensionality, we know that is a closed set in , which implies that necessarily . Finally, as a consequence of Lemma 5, the nontrivial intersection of the ranges is equivalent to the operator achieving its maximum norm. □
Observe that Corollary 4 follows directly from Corollary 5, since every normal operator is, specifically, normaloid. Accordingly, if , , and T belongs to any of the following subclasses—namely, self-adjoint, normal, or normaloid operators (see (2))—then we have
A partir de lo anterior, it is natural to wonder whether the fact that a nonzero operator satisfies the equality implies that T is normaloid. The following counterexample demonstrates that this is not necessarily the case.
On the other hand, it is well known that every normaloid operator is also spectraloid, i.e., , which naturally raises the question of whether this new class of operators satisfies this equality. We provide two examples: the first shows that there exist spectraloid operators that are not normaloid yet satisfy the norm equality studied in this manuscript, while the second gives a negative answer to the previous question.
Example 3.
Motivated by the manuscript of Goldberg and Zwas [], we consider, for each , the following block matrix
where denotes the identity matrix.
In [], the authors showed that is a spectraloid matrix but not normaloid (that is, the inclusion between these two subclasses is strict). Moreover, for each , it holds that
By performing block-wise multiplication, we obtain
Therefore,
for any .
Another potential counterexample, based on Proposition 1, is to exhibit a nonzero operator that is EP but not normaloid.
Example 4.
Consider the matrix
It is immediate to see that and that if we take , then
hence
which shows that T is not spectraloid. Furthermore, it can be readily verified that
thus concluding that T is not normaloid.
Since T is invertible, its Moore–Penrose pseudoinverse coincides with its usual inverse. In particular, we have which shows that T is an EP operator, and
3. Conclusions
In this paper, we investigated the operator norm of the sum of two orthogonal projections on a complex Hilbert space, identifying necessary and sufficient conditions for this norm to attain its maximum value, namely 2. Our analysis was guided by geometric considerations, specifically the Dixmier angle between closed subspaces, and we expressed our conditions directly in terms of the ranges of the involved projections.
As a central application, we extended these results to bounded linear operators with closed range through their Moore–Penrose inverses. In particular, we derived sufficient conditions under which the operator
has norm equal to 2. This result generalizes previous findings from the finite-dimensional setting to the broader context of general (possibly infinite-dimensional) Hilbert spaces.
Beyond these core contributions, our approach highlights deep connections between subspace geometry, projection theory, and the theory of generalized inverses. The interplay between these areas opens several potential avenues for future research. For instance, the further exploration of norm-attaining properties for sums or compositions of more than two projections, or for other classes of generalized inverses, could yield new insights.
We believe that the techniques and results developed in this work may serve as a foundation for a broader line of research in operator theory and Hilbert space geometry. In particular, this study may be a starting point for new investigations into norm inequalities, operator angles, and their role in solving structural problems related to projections and closed-range operators.
Author Contributions
Methodology, C.C., K.F., S.A. and S.F.; Validation, C.C., K.F., S.A. and S.F.; Formal analysis, C.C., K.F., S.A. and S.F.; Writing—original draft, C.C., K.F., S.A. and S.F.; Writing—review & editing, C.C., K.F., S.A. and S.F.; Supervision, C.C., K.F., S.A. and S.F.; Funding acquisition, C.C., K.F., S.A. and S.F. All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.
Funding
Researchers Supporting Project number (PNURSP2025R514), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Data Availability Statement
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.
Acknowledgments
The authors would like to extend their sincere appreciation to the anonymous referees for their invaluable comments and suggestions, which greatly contributed to the enhancement of our article. Additionally, the first author would like to acknowledge the support received from Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2025R514), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Conflicts of Interest
The authors declare that they have no competing interests.
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