Invariant submodules of modular operators and Lomonosov type theorem for Hilbert C*-modules

Kamran Sharifi

REVIEW 3 major objections 5 minor 35 references

Invariant submodules of modular operators and Lomonosov type theorem for Hilbert C*-modules

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read On Hilbert C*-modules over finite-dimensional C*-algebras, every nonzero compact operator on a non-finitely-generated module has a proper nonzero hyperinvariant submodule.

desk verdict Natural extension of Lomonosov to Hilbert C*-modules over finite-dimensional C*-algebras; the main theorem is likely correct but the proof has repairable gaps in two key lemmas. read the letter →

arxiv 2506.01161 v1 pith:YSUOKHIH submitted 2025-06-01 math.OA math.FA

Kamran Sharifi This is my paper

classification math.OAmath.FA MSC 46L0847A0546C5046L05

keywords HilbertC*-moduleinvariantsubmodulehyperinvariantcompactoperatorLomonosovtheoremfinite-dimensionalC*-algebrageneralizedinverseequation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Invariant submodules are the Hilbert $C^*$-module analogue of invariant subspaces: closed submodules preserved by an adjointable operator. The paper develops their basic theory, connects them to projections, reducing submodules, and the operator equation $XTX = TX$, and then proves a Lomonosov-type theorem. The theorem states that if $A$ is a finite-dimensional $C^*$-algebra and $E$ is a Hilbert $A$-module that is not finitely generated, then every nonzero compact operator on $E$ has a proper nonzero hyperinvariant submodule. This extends a classical result from Hilbert and Banach spaces to a module setting where self-duality and orthogonal complementation fail in general, so the proof has to build the needed compactness by hand.

What carries the argument

The mechanism that carries the argument is a weak compactness property special to Hilbert modules over finite-dimensional $C^*$-algebras (Lemma 4.1, taken from [1, Theorem 2.3] and [5, Proposition 2.1]). It says that every bounded sequence has a subsequence whose images under every compact operator converge in norm, and it follows from the self-duality of such modules proved by Frank [10, Proposition 4.4]. That property makes the range of $I - K$ an orthogonal summand (Lemma 4.3), turns nonzero spectral points into eigenvalues (Lemma 4.6), and finally makes the closed image $KB$ of the unit ball compact enough for the Lomonosov covering argument. The covering argument itself, with the constant $c = \max \|S_i\|$ and the vanishing of $\|(cK)^m\|$, is the same combinatorial skeleton as in the classical proof.

What would settle it

Concrete check: try to construct a non-finitely-generated Hilbert module over a finite-dimensional $C^*$-algebra and a nonzero compact operator on it with either a nonzero spectral point that is not an eigenvalue, or a vector whose orbit under the commutant is dense while the spectral-radius estimate (4.3) still holds; either observation would falsify Theorem 4.8 or one of the lemmas that feed it.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.8: for a finite-dimensional $C^*$-algebra $A$ and a Hilbert $A$-module $E$ that is not finitely generated, every nonzero compact operator $K$ on $E$ admits a proper nonzero hyperinvariant submodule---a closed submodule invariant under every $S \in L(E)$ that commutes with $K$. If $K$ has a nonzero spectral value $\lambda$, then $\mathrm{Ker}(\lambda I - K)$ is such a submodule by Lemma 4.6, and it is proper because $K$ is compact and $E$ is not finitely generated. If the spectrum of $K$ is just $\{0\}$, the paper runs the Lomonosov covering argument: the sets $O(S) = \{u : \|Su - x_0\| < 1\}$, for $S$ commuting with $K$, cover the compact set $KB$, and the spectral-radius condition (4.3) forces a contradiction with $0$ lying outside the closed unit ball $B$ centered at $x_0$. The upshot is that the closed submodule generated by the orbit of some vector under the commutant of $K$ is neither zero nor all of $E$.

Load-bearing premise

The load-bearing premise is Lemma 4.1: for Hilbert modules over finite-dimensional $C^*$-algebras, every bounded sequence has a subsequence whose images under every compact operator converge in norm; if that weak compactness ever failed, the range-closedness, spectral-eigenvalue, and covering arguments in Theorem 4.8 would all break down.

Editorial extensions

If this is right

Corollary 4.9: any adjointable operator $S$ commuting with a nonzero compact operator on such a module has a proper nonzero invariant submodule.
Over $C^*$-algebras of compact operators, the equivalence of having a nontrivial invariant submodule, having a nontrivial projection $P$ with $PTP = TP$, and having a nontrivial solution of $STS = TS$ becomes unconditional because every closed submodule is complemented (Corollary 3.1).
For a complemented invariant submodule $W$ with closed range of $TP_W$, the solutions of $XTX = TX$ are exactly $P_{\mathrm{Ran}(TP_W)} + P_W Z(I - P_{\mathrm{Ran}(TP_W)})$ with arbitrary $Z \in L(E)$ (Corollaries 2.6 and 3.2).
A complemented submodule that reduces a Moore-Penrose invertible operator $T$ also reduces its Moore-Penrose inverse $T^\dagger$ whenever $0$ is outside the $C^*$-numerical range of $T$ (Theorem 2.12).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: the same hyperinvariant-submodule conclusion plausibly holds for any adjointable operator whose commutant contains a nonzero compact operator on these modules, mirroring Lomonosov's stronger formulation in Banach spaces.
Inference: because the proof keys off self-duality, a natural test case is Hilbert modules over finite-dimensional $W^*$-algebras, where a version of the weak compactness property might persist under extra hypotheses.
Inference: a constructive version of the theorem would specify the vector whose commutant orbit yields the hyperinvariant submodule in the $\sigma(K) = \{0\}$ case; working it out for $E = \ell^2 \otimes M_n(\mathbb{C})$ would give an explicit, checkable example.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read on Sharifi's paper. The main claim, Theorem 4.8, is a Lomonosov-type theorem for compact operators on Hilbert C*-modules over finite-dimensional C*-algebras, when the module is not finitely generated. That is a real extension of Lomonosov's 1973 result to a setting where it hadn't been done, and the proof strategy is a faithful adaptation of Lomonosov's covering argument. The reader checked the literature: the result is not in the cited references, there's no circularity, and no fitted parameters. If the theorem is correct—and I think it is—it's a solid contribution to Hilbert C*-module theory.

What the paper does well: it gives a clean statement and proof of the extension, and the surrounding sections on invariant submodules, operator equations, and generalized inverses are mostly standard but competently handled. The proof of Theorem 4.8 itself, once you accept Lemma 4.1 and Lemma 4.6, is a direct and readable version of the classical argument.

Now the soft spots, in proportion. Lemma 4.1, which is imported from [1,5] and is the compactness engine, has a proof sketch in the paper that doesn't work as written. The text just picks, for each v, a subsequence of ⟨v,·⟩; there's no diagonalization to get a single subsequence valid for all v, and the limit map g(v) isn't shown to be well-defined or A-linear. The stress-test note is right about this: the lemma is true (E is reflexive, so Eberlein–Smulian would do it, or the cited references likely have the full proof), but the proof gap is real. Lemma 4.6 also asserts without argument that the ranges of L^n form a strictly decreasing sequence when L is injective but not surjective; that needs a proof, and the contradiction argument depends on it. Minor issues: Lemma 2.2(2) is unused and its hypothesis is vague, and Lemma 4.3 silently assumes proximinality of closed submodules for the distance-minimizing sequence. None of this undermines the truth of Theorem 4.8; it's all repairable.

Who this is for: researchers in Hilbert C*-module theory and operator theory who care about invariant subspaces in module settings. A referee should be sent in, and should require a correct proof of Lemma 4.1 and a justification for the strict decrease in Lemma 4.6. I would accept this for review, not desk-reject.

Referee Report

3 major / 5 minor

Summary. This paper develops a theory of invariant and reducing submodules for adjointable operators on Hilbert C*-modules. The early sections define invariant submodules, characterize them via projections satisfying PTP=TP and via nontrivial solutions of STS=TS, derive block-matrix decompositions relative to complemented submodules, and study the Moore-Penrose inverse under a numerical-range condition. The later sections specialize to Hilbert modules over C*-algebras of compact operators and over finite-dimensional C*-algebras. The main result, Theorem 4.8, asserts that if A is finite-dimensional and E is a non-finitely generated Hilbert A-module, then every nonzero compact operator on E has a proper nonzero hyperinvariant submodule; Corollary 4.9 draws the usual Lomonosov consequence for operators commuting with compact operators. The proof follows Lomonosov's spectral and covering argument, using a compactness lemma (Lemma 4.1) imported from [1,5] and a spectral lemma (Lemma 4.6).

Significance. Should Theorem 4.8 hold, it is a substantive extension of Lomonosov's theorem to a class of Hilbert C*-modules, and the framework of invariant submodules introduced here may be useful for further module operator theory. The paper gives explicit formulas for solution sets of operator equations, which are concrete and potentially applicable. The main proof is a recognizable Lomonosov argument and, modulo the gaps identified below, is likely correct. However, the two load-bearing lemmas in Section 4 are not fully proved as written: Lemma 4.1 has an invalid proof sketch, and Lemma 4.6 contains an asserted strict descent that is not justified. These are repairable, but until then the central theorem is not established by the manuscript. The paper does not ship machine-checked proofs or code; its value rests on the mathematical argument and on the correctness of the imported compactness results.

major comments (3)

[Section 4, Lemma 4.1] The proof as written is not a valid proof of the statement. For each v in E, the argument produces a subsequence (zeta_{n_k(v)}) for which <v, zeta_{n_k(v)}> converges, but these subsequences need not be compatible, so the limit map g(v) is not defined on a common subsequence and need not be A-linear. Since (4.1) and (4.2) are subsequently used with a single subsequence in Lemma 4.3, in Lemma 4.6, and in the covering argument of Theorem 4.8, this is a load-bearing gap. The lemma may be correct (for example, via reflexivity of E and the Eberlein-Smulian theorem), and [1, Theorem 2.3] and [5, Proposition 2.1] may supply it, but the paper must either prove it fully or import it with a correct self-contained argument.
[Section 4, Lemma 4.6] The assertion that the sequence F_1 contains F_2 contains ... is strictly decreasing is not proved. From injectivity of L and Lemma 4.3 one knows that each F_n is closed, but strictness requires an additional cancellation argument: if Ran(L^n)=Ran(L^{n-1}), then for every y in E there is z with L^{n-1}y = L^n z = L^{n-1}(Lz), and injectivity of L^{n-1} gives y=Lz, contradicting properness of Ran(L). The proof should also state that L^n = I - K_n with K_n compact, so that Lemma 4.3 applies to each F_n. Without strictness, the selection of f_n with ||f_n - F_{n+1}|| > 1/2 has no basis, and the displayed contradiction in the proof of part (2) collapses.
[Section 4, Lemma 4.3] The proof asserts the existence of u_n in Ker(L) achieving the distance from x_n to Ker(L). This existence is true because Hilbert modules over finite-dimensional C*-algebras are reflexive Banach spaces, but the text does not justify it. Since this proximinality is used to form the sequence zeta_n, a short proof or explicit citation should be supplied. This is a smaller gap than the two above, but it is still a step that a reader cannot verify from the text as written.

minor comments (5)

[Abstract] The word 'underling' should be 'underlying'.
[Section 1, Introduction] The phrase 'In the early thirties' should read 'In the early 1930s'; the historical comments on the invariant subspace problem would also benefit from a clear statement that the Enflo and Neville preprints have not yet been peer-reviewed.
[Section 2, proof of Lemma 2.3] In the second part of the proof, 'suppose T S=0' is a typographical slip for 'suppose TS=0'; the surrounding argument does not otherwise define the first case clearly.
[Section 4, proof of Theorem 4.8] The notation 'KB=KB is a compact subset' is confusing; the closure of K(B) should be denoted overline{K(B)} or another explicit symbol to distinguish the set from its closure.
[Problem 2.16] The statement contains a formatting error: 'T=I. a' should be typeset as 'T = I \cdot a'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 4.8 is derived from imported background results and direct algebraic arguments; self-citations are contextual, not load-bearing.

full rationale

The paper's central derivation chain is not circular. Section 2 introduces invariant submodules and proves reformulations by direct definitions (Theorem 2.4, Proposition 2.7, Proposition 2.11); these are equivalences, not fitted predictions. Section 4's Lomonosov-type theorem rests on Lemma 4.1, which is attributed to [1, Theorem 2.3] and [5, Proposition 2.1] and on Frank's self-duality result [10, Proposition 4.4]; these are independent external inputs, not restatements of the target theorem. Lemma 4.6 and Lemma 4.3 use standard compactness and Riesz-lemma arguments; no parameter is fitted and no quantity called a prediction is defined in terms of the conclusion. The author's self-citations ([13,14,30,31,32]) appear in the preliminary definition of partial isometries and in Section 3's contextual characterization of C*-algebras of compact operators; none of these is needed to prove Theorem 4.8, so they are not load-bearing circularity. A skeptical reader could question the proof sketch of Lemma 4.1: the text chooses a convergent subsequence of <v, zeta_n> for each v and then invokes self-duality, without explicitly showing that one common subsequence works for all v. This is a potential proof gap or an over-compressed import from [1,5], not a circularity, because the lemma is cited from independent sources and is not equivalent to the invariant-submodule conclusion. Therefore the derivation is self-contained relative to standard background results and receives a low circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters and no new postulated entities. It relies on standard Hilbert C*-module theory plus two substantive inputs from the literature: complementedness of closed submodules over finite-dimensional C*-algebras and the weak compactness property (Lemma 4.1). These are domain assumptions, not ad hoc constructions.

assumptions (6)

standard math Standard theory of Hilbert C*-modules: closed-range adjointable operators give orthogonal decompositions E=Ker(T)⊕Ran(T*) and F=Ran(T)⊕Ker(T*).
Invoked throughout, e.g., Section 1 and Lemma 4.3, from [20, Theorem 3.2].
domain assumption Every closed submodule of a Hilbert module over a C*-algebra of compact operators (in particular, over a finite-dimensional C*-algebra) is orthogonally complemented.
Used in Corollaries 3.1-3.2 and in Lemma 4.3 to guarantee projections and closest points; cited to [4,29] in Section 3.
domain assumption Self-duality of Hilbert C*-modules over finite-dimensional C*-algebras, used to prove Lemma 4.1.
Lemma 4.1 depends on self-duality to lift a bounded module map g to an element ζ; cited to [10, Proposition 4.4] in the proof.
domain assumption Weak compactness lemma (Lemma 4.1) as proven in [1] and [5].
This is the key imported result that makes the Lomonosov argument work in the module setting; it is assumed as a black box.
standard math For an injective linear operator L, if L^n(E)=L^{n-1}(E) for some n then L is surjective.
Needed to prove the strict decrease of ranges F_n in Lemma 4.6; not stated explicitly in the paper.
standard math Spectral radius formula: if σ(K)={0} then lim ||K^n||^{1/n}=0.
Used in Theorem 4.8 to make x_m converge to 0; standard Banach algebra result.

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Cite this review

Pith. "Pith review of Invariant submodules of modular operators and Lomonosov type theorem for Hilbert C*-modules." pith.science (2026). https://pith.science/paper/YSUOKHIH

@misc{pith2026250601161,
  author       = {Pith},
  title        = {Pith review of: Invariant submodules of modular operators and Lomonosov type theorem for Hilbert C*-modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YSUOKHIH}},
  note         = {Machine review of arXiv:2506.01161}
}

read the original abstract

In this paper, we introduce the notion of invariant submodule in the theory of Hilbert C*-modules and study some basic properties of bounded adjointable operators and their generalized inverses which have nontrivial invariant submodules. We demonstrate the representation of the solution set of an operator equation on Hilbert C*-modules by taking advantage of invariant submodules. In particular, we consider the special cases of finite dimensional C*-algebras and C*-algebras of compact operators as the underling C*-algebra to simplify our results, and obtain a Lomonosov type theorem for compact operators on some Hilbert C*-modules.

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Reference graph

Works this paper leans on

35 extracted references · 33 canonical work pages

[1]

Arambaˇ si´ c, D

Lj. Arambaˇ si´ c, D. Baki´ c, and R. Raji´ c, Finite-dimensional Hilbert C*-modules,Banach J. Math. Anal. 4(2010), 147-157

work page 2010
[2]

Aronszajn and K

N. Aronszajn and K. T. Smith, Invariant subspaces of completely continuous operatorsAnnals of Math- ematics,60(1954), 345-350

work page 1954
[3]

Arveson,An Invitation toC ∗-algebras, Springer, New York, 1976

W. Arveson,An Invitation toC ∗-algebras, Springer, New York, 1976

work page 1976
[4]

Baki´ c and B

D. Baki´ c and B. Guljaˇ s, Hilbert C*-modules over C*-algebras of compact operators,Acta Sci. Math. (Szeged)68(2002), no. 1-2, 249-269

work page 2002
[5]

Chmieli´ nski, D

J. Chmieli´ nski, D. Iliˇ sevi´ c, M. S. Moslehian and Gh. Sadeghi, Perturbation of the Wigner equation in inner product C*-modules,J. Math. Phys.49(2008), no. 3, 033519, 8 pp

work page 2008
[6]

J. B. Conway, A course in functional analysis, Graduate Texts in Mathematics 96, Springer 1990

work page 1990
[7]

D. S. Cvetkovi´ c-Ili´ c, Solvability and different solutions of the operator equationXAX=BX,Ann. Funct. Anal.14(2023), no. 1, Paper No. 5

work page 2023
[8]

P. H. Enflo, On the invariant subspace problem in Hilbert spaces, arXiv.2305.15442

arXiv

Show all 35 references

[9]

X. Fang, M. S. Moslehian and Q. Xu, On majorization and range inclusion of operators on Hilbert C*-modules,Linear Multilinear Algebra66(2018), no. 12, 2493-2500

2018
[10]

Frank, Self-duality and C*-reflexivity of Hilbert C*-modules,Z

M. Frank, Self-duality and C*-reflexivity of Hilbert C*-modules,Z. Anal. Anwendungen9(1990), 165- 176

1990
[11]

Frank, Geometrical aspects of Hilbert C*-modules,Positivity3(1999), 215-243

M. Frank, Geometrical aspects of Hilbert C*-modules,Positivity3(1999), 215-243

1999
[12]

Frank, Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules,J

M. Frank, Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules,J. K-Theory2(2008), 453-462

2008
[13]

Frank and K

M. Frank and K. Sharifi, Adjointability of densely defined closed operators and the Magajna-Schweizer theorem,J. Operator Theory63(2010), 271-282

2010
[14]

Frank and K

M. Frank and K. Sharifi, Generalized inverses and polar decomposition of unbounded regular operators on Hilbert C*-modules,J. Operator Theory64(2010), 377-386

2010
[15]

Guljaˇ s, Unbounded operators on Hilbert C*-modules over C*-algebras of compact operators,J

B. Guljaˇ s, Unbounded operators on Hilbert C*-modules over C*-algebras of compact operators,J. Operato Theory59(2008), no. 1, 179-192

2008
[16]

Holbrook, E

J. Holbrook, E. Nordren, H. Radjavi and P. Rosenthal, On the operator equationAX=XAX,Linear Algebra Appl.295(1999), no. 1-3, 113-116. 18 K. SHARIFI

1999
[17]

Kaplansky, Module over operator algebra,Amer

I. Kaplansky, Module over operator algebra,Amer. J. Math,75(1953), 839-858

1953
[18]

E. G. Katsoulis, The reflexive closure of the adjointable operators,Illinois J. Math.58(2014), no. 2, 359-367

2014
[19]

C. S. Kubrusly,An Introduction to Models and Decompositions in Operator Theory, Birkh¨ auser, Boston 1997

1997
[20]

E. C. Lance,Hilbert C*-Modules, LMS Lecture Note Series 210, Cambridge Univ. Press, 1995

1995
[21]

N. Liu, W. Luo and Q. Xu, The polar decomposition for adjointable operators on Hilbert C*-modules and centered operators,Adv. Oper. Theory3(2018), no. 4, 855-867

2018
[22]

V. I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely con- tinuous operator. (Russian)Funkcional. Anal. i Priloˇ zen.7(1973), no. 3, 55-56

1973
[23]

V. M. Manuilov, Diagonalization of compact operators in Hilbert modules over finite W*-algebras,Ann. Global Anal. Geom.13(1995), no. 3, 207-226

1995
[24]

V. M. Manuilov and E. V. Troitsky,Hilbert C*-modules, Translated from the 2001 Russian original by the authors, Translations of Mathematical Monographs, 226. American Mathematical Society, Providence, RI, 2005

2001
[25]

C. W. Neville, A proof of the invariant subspace conjecture for separable Hilbert spaces, arXiv.2307.08176

arXiv
[26]

W. L. Paschke, Inner product modules overB ∗-algebras,Trans Amer. Math. Soc.182(1973), 443-468

1973
[27]

P. W. Prasetyo and D. A. Yuwaningsih,On the existing of fully invariant submodule, Journal of Physics: Conference Series. Vol. 1188. No. 1. IOP Publishing, 2019

2019
[28]

Radjavi and P

H. Radjavi and P. Rosenthal,Invariant subspaces, Second edition. Dover Publications, Inc., Mineola, NY, 2003

2003
[29]

Schweizer, A description of Hilbert C*-modules in which all closed submodules are orthogonally closed,Proc

J. Schweizer, A description of Hilbert C*-modules in which all closed submodules are orthogonally closed,Proc. Amer. Math. Soc.127(1999), 2123-2125

1999
[30]

Sharifi, Descriptions of partial isometries on Hilbert C*-modules,Linear Algebra Appl.4 ¯31 (2009), 883-887

K. Sharifi, Descriptions of partial isometries on Hilbert C*-modules,Linear Algebra Appl.4 ¯31 (2009), 883-887

2009
[31]

Sharifi, The product of operators with closed range in Hilbert C*-modules,Linear Algebra Appl

K. Sharifi, The product of operators with closed range in Hilbert C*-modules,Linear Algebra Appl. 435(2011), 1122-1130

2011
[32]

Sharifi, Generic properties of module maps and characterizing inverse limits of C*-algebras of compact operators,Bull

K. Sharifi, Generic properties of module maps and characterizing inverse limits of C*-algebras of compact operators,Bull. Malays. Math. Sci. Soc.(2)36(2013), no. 2, 481-489

2013
[33]

N. E. Wegge-Olsen,K-theory andC ∗-algebras: a Friendly Approach, Oxford University Press, Oxford, England, 1993

1993
[34]

Xu and L

Q. Xu and L. Sheng, Positive semi-definite matrices of adjointable operators on Hilbert C*-modules, Linear Algebra Appl.428(2008), 992-1000

2008
[35]

Zhang, The characterization of Moore-Penrose inverse module maps and their continuity,Rocky Mountain J

L. Zhang, The characterization of Moore-Penrose inverse module maps and their continuity,Rocky Mountain J. Math.38(2008), no. 1, 351-357. INV ARIANT SUBMODULES OF MODULAR OPERATORS 19 Kamran Sharifi, F aculty of Mathematical Sciences, Shahrood University of Technology, P. O. B...

2008

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Arambaˇ si´ c, D

Lj. Arambaˇ si´ c, D. Baki´ c, and R. Raji´ c, Finite-dimensional Hilbert C*-modules,Banach J. Math. Anal. 4(2010), 147-157

work page 2010

[2] [2]

Aronszajn and K

N. Aronszajn and K. T. Smith, Invariant subspaces of completely continuous operatorsAnnals of Math- ematics,60(1954), 345-350

work page 1954

[3] [3]

Arveson,An Invitation toC ∗-algebras, Springer, New York, 1976

W. Arveson,An Invitation toC ∗-algebras, Springer, New York, 1976

work page 1976

[4] [4]

Baki´ c and B

D. Baki´ c and B. Guljaˇ s, Hilbert C*-modules over C*-algebras of compact operators,Acta Sci. Math. (Szeged)68(2002), no. 1-2, 249-269

work page 2002

[5] [5]

Chmieli´ nski, D

J. Chmieli´ nski, D. Iliˇ sevi´ c, M. S. Moslehian and Gh. Sadeghi, Perturbation of the Wigner equation in inner product C*-modules,J. Math. Phys.49(2008), no. 3, 033519, 8 pp

work page 2008

[6] [6]

J. B. Conway, A course in functional analysis, Graduate Texts in Mathematics 96, Springer 1990

work page 1990

[7] [7]

D. S. Cvetkovi´ c-Ili´ c, Solvability and different solutions of the operator equationXAX=BX,Ann. Funct. Anal.14(2023), no. 1, Paper No. 5

work page 2023

[8] [8]

P. H. Enflo, On the invariant subspace problem in Hilbert spaces, arXiv.2305.15442

arXiv