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Vol. 1, vol. 48. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence (2000)\nBessaga, C., Pełczyński, A.: Selected Topics in Infinite-dimensional Topology. Polska Akademia Nauk. Instytut Matematyczny, Monografie matematyczne, Warsaw (1975)\nBliedtner, J., Loeb, P.A.: The optimal differentiation basis and liftings of \\(L^\\infty \\). Trans. Am. Math. Soc. 352, 4693–4710 (2000)\nBochner, S.: Integration von Funktionen, deren Werte die Elemente eines Vektorraumes sind. Fundam. Math. 20, 262–276 (1933)\nBogachev, V.I.: Measure Theory, vol. II. Springer, Berlin (2007)\nBourbaki, N.: Topologie générale. Hermann, Paris (1965)\nBrena, C., Gigli, N.: Local vector measures. Preprint. arXiv:2206.14864 (2022)\nBruckner, A.M.: Differentiation of integrals. Am. Math. Mon. 78, 53 (1971)\nBruckner, A.M., Bruckner, J.B., Thomson, B.S.: Real Analysis. Prentice-Hall, Upper Saddle River (1997)\nBruckner, A.M., Bruckner, J.B., Thomson, B.S.: Real Analysis, 2nd edn. ClassicalRealAnalysis.com, Vancouver (2008)\nDi Marino, S., Lučić, D., Pasqualetto, E.: Representation theorems for normed modules. Preprint. arXiv:2109.03509 (2021)\nDiestel, J., Uhl, J.J., Jr.: Vector Measures. American Mathematical Society, Providence (1977)\nDunford, N.: Integration in general analysis. Trans. Am. Math. Soc. 37, 441–453 (1935)\nDunford, N., Schwartz, J.: Linear Operators. (I): General Theory. Interscience Publishers, Pure and Applied Mathematics, New York (1958)\nFremlin, D.: Measure Theory: Measure Algebras. Measure Theory, vol. 3. Torres Fremlin, Colchester (2011)\nGigli, N., Lučić, D., Pasqualetto, E.: Duals and pullbacks of normed modules. Preprint. arXiv:2207.04972 (2022)\nGigli, N.: Lecture notes on differential calculus on \\({\\sf RCD}\\) spaces. Publ. Res. Inst. Math. Sci. 54, 855–918 (2018)\nGigli, N.: Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below. Mem. Am. Math. Soc. 251, 161 (2018)\nGraf, S., von Weizsäcker, H.: On the Existence of Lower Densities in Noncomplete Measure Spaces. Measure Theory, pp. 155–158. Springer, Berlin (1976)\nGuo, T.X.: Recent progress in random metric theory and its applications to conditional risk measures. Sci. China Math. 54, 633–660 (2011)\nGutman, A.E.: Banach bundles in the theory of lattice-normed spaces. I. Continuous Banach bundles. Sib. Adv. Math. 3, 1–55 (1993)\nGutman, A.E.: Banach bundles in the theory of lattice-normed spaces. II. Measurable Banach bundles. Sib. Adv. Math. 3, 8–40 (1993)\nGutman, A.E., Koptev, A.V.: Dual Banach Bundles. Nonstandard Analysis and Vector Lattices, Mathematical Applications, vol. 525, pp. 105–159. Kluwer Academic Publishers, Dordrecht (2000)\nHaydon, R., Levy, M., Raynaud, Y.: Randomly Normed Spaces. Travaux en Cours [Works in Progress], vol. 41. Hermann, Paris (1991)\nHeinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001)\nHoffmann-Jørgensen, J.: Existence of conditional probabilities. Math. Scand. 28, 257–264 (1971)\nHytönen, T., Neerven, J., Veraar, M., Weis, L.: Analysis in Banach Spaces: Volume I: Martingales and Littlewood-Paley Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge\u002FA Series of Modern Surveys in Mathematics, Springer (2016)\nJohnson, R.A.: Atomic and nonatomic measures. Proc. Am. Math. Soc. 25, 650–655 (1970)\nJohnson, R.A.: Strong liftings that are not Borel liftings. Proc. Am. Math. Soc. 80, 234–236 (1980)\nKölzow, D.: Differentiation von Maßen. Lecture Notes in Mathematics. Springer, Berlin (1968)\nLebesgue, H.: Sur l’intégration des fonctions discontinues. Ann. Sci. Éc. Norm. Supér. 3(27), 361–450 (1910)\nLoeb, P.A., Talvila, E.: Lusin’s theorem and Bochner integration. Sci. Math. Jpn. 10, 55–62 (2004)\nLosert, V.: A measure space without the strong lifting property. Math. Ann. 239, 119–128 (1979)\nLučić, D., Pasqualetto, E.: The Serre–Swan theorem for normed modules. Rend. Circ. Mat. Palermo 2(68), 385–404 (2019)\nLukeš, J., Malý, J., Zajíček, L.: Fine Topological Methods in Real Analysis and Potential Theory. Vol. 1189 of Lecture Notes in Mathematics. Springer, Berlin (1986)\nMaharam, D.: On a theorem of von Neumann. Proc. Am. Math. Soc. 9, 987–994 (1958)\nMokobodzki, G.: Relévement borélien compatible avec une classe d’ensembles négligeables. Application á la désintégration des mesures. Séminaire de probabilités de Strasbourg 9, 437–442 (1975)\nShelah, S.: Lifting problem of the measure algebra. Isr. J. Math. 45, 90–96 (1983)\nStrauss, W., Macheras, N.D., Musiał, K.: Liftings. Handbook of Measure Theory: In two volumes (edited by E. Pap), Volume II, Part 7. Elsevier Science, Amsterdam (2002)\nTraynor, T.: An elementary proof of the lifting theorem. Pac. J. Math. 53, 267–272 (1974)\nTulcea, A.I., Tulcea, C.I.: On the existence of a lifting commuting with the left translations of an arbitrary locally compact group. In: Proceedings of the Fifth Berkeley Symposium of Mathematical Statistics and Probability (Berkeley, Calif., 1965\u002F66), Vol. II: Contributions to Probability Theory, Part 1, pp. 63–97. University of California Press (1967)\nTulcea, C.I.: On the lifting property and disintegration of measures. Bull. Am. Math. Soc. 71, 829–842 (1965)\nTulcea, A.I., Tulcea, C.I.: On the lifting property (I). J. Math. Anal. Appl. 3, 537–546 (1961)\nTulcea, A.I., Tulcea, C.I.: Liftings for abstract valued functions and separable stochastic processes. Probab. Theory Relat. Fields 13, 114–118 (1969)\nTulcea, A.I., Tulcea, C.I.: Topics in the Theory of Lifting. Vol. 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1969)\nvon Neumann, J.: Algebraische Repräsentanten der Funktionen “bis auf eine Menge von Maße Null’’. J. 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Appl.; citation_title=Characterization of shift-invariant spaces on a class of nilpotent Lie groups with applications; citation_author=B Currey, A Mayeli, V Oussa; citation_volume=20; citation_issue=2; citation_publication_date=2014; citation_pages=384-400; citation_doi=10.1007\u002Fs00041-013-9316-z; citation_id=CR10\ncitation_journal_title=J. Math. Anal. Appl.; citation_title=The structure of shift-invariant spaces on locally compact abelian group; citation_author=RA Kamyabi Gol, R Raisi Tousi; citation_volume=340; citation_publication_date=2008; citation_pages=219-225; citation_doi=10.1016\u002Fj.jmaa.2007.08.039; citation_id=CR11\ncitation_journal_title=Can. J. Math.; citation_title=Frames and stable bases for shift-invariant subspaces of \n                    \n                    \n                    \n                  ; citation_author=A Ron, Z Shen; citation_volume=47; citation_issue=5; citation_publication_date=1995; citation_pages=1051-1094; citation_doi=10.4153\u002FCJM-1995-056-1; citation_id=CR12\ncitation_journal_title=Linear Multilinear Algebra; citation_title=Translation generated oblique dual frames on locally compact groups; citation_author=S Sarkar, NK Shukla; citation_publication_date=2023; citation_doi=10.1080\u002F03081087.2023.2173718; citation_id=CR13",{"EN":229},"In the context of a connected, simply connected nilpotent Lie group, whose representations are square-integrable modulo the center, we find characterization results of extra-invariant spaces under the left translations associated with the range functions. 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By studying the characterisation of two idempotents generated from factorizations of idempotent operator, we give a necessary and sufficient condition for when there is no another idempotent operator \n                \n                  \n                \n                $$\\Pi _3$$\n                \n               which has the same range with \n                \n                  \n                \n                $$\\Pi _1\\Pi _2$$\n                \n               such that \n                \n                  \n                \n                $$\\Pi _1+\\Pi _2-\\Pi _3$$\n                \n               is idempotent.",{"EN":313},"Factorizations of idempotent operator as products of two 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Izvestya AN KazSSR, ser. fiz.-mat. 3, 47–51 (1982)\nAkishev, G.A.: On degrees of approximation of some classes by polynomials with respect to generalized Haar system. Sib. Electron. Math. Rep. 3, 92–105 (2006)\nAkishev, G.: On the orders \\(M\\)-terms approximations of classes of functions of the symmetrical space. Mat. Zh. 14(4), 44–71 (2014)\nAkishev, G.: Estimates for best approximations of functions from the logarthmic smoothness class in the Lorentz space. Trudy Instituta Matematiki i Mekhaniki UrO RAN. 23(3), 3–21 (2017)\nAkishev, G.: An inequality of different metrics in the generalized Lorentz space. Trudy Instituta Matematiki i Mekhaniki UrO RAN 24(4), 5–18 (2018)\nAkishev, G.: On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz spaces. Trudy Instituta Matematiki i Mekhaniki UrO RAN 25(2), 9–20 (2019)\nBesov, O.V.: Investigation of a class of function spaces in connection with imbedding and extension theorems. Tr. Mat. Inst. Steklov. 60, 42–81 (1961)\nBurenkov V.I.: Imbedding and extension theorems for classes of differentiable functions of several variables defined on the entire spaces, In Itogi Nauki i Tekhniki. Seriya “Matematicheskii Analiz“ pp.71-155, Moscow (1966)\nCobos, F., Dominguez, O.: On Besov spaces of logarithmic smoothness and Lipschitz spaces. J. Math. Anal. Appl. 425, 71–84 (2015)\nCobos, F., Milman, M.: On a limit class of approximation spaces. Numer. Funct. Anal. Optimiz. 11, 11–31 (1990)\nDeVore, R.A., Riemenschneider, S.D., Sharpley, R.C.: Weak interpolation in Banach spaces. J. Funct. Anal. 33, 58–94 (1979)\nDitzian, Z., Tikhonov, S.: Ul’yanov and Nikol’skii-type inequalities. J. Approx. Theory 133(1), 100–133 (2005)\nDominguez, O., Tikhonov, S.: Function spaces of logarithmic smoothness: embeding and characterizations. Preprint (2018). arXiv:1811.06399v [math.FA]. Mem. Amer. Math. Soc. (accepted)\nDung, D., Temlyakov, V., Ullrich, T.: Hyperbolic cross approximation. Adv. Courses Math, CRM Barselona (2018)\nDzhafarov, A.S.: Embedding theorems for classes of functions with differential properties in the norms of special spaces. Dokl. AN Azerb. SSR. 21(2), 10–14 (1965)\nGol’dman, M.L.: On the inclusion of generalized Hölder classes. Math. Notes. 12(3), 626–631 (1972)\nJanson, S.: On the interpolation of sublinear operators. Stud. Math. 75, 51–53 (1982)\nKashin B.S., Temlyakov V.: On a norm and approximation characteristics of classes of functions of several variables. Metric theory of functions and related problems in analysis (Russian). Izd. Nauchno-Issled. Aktuarno-Finans. Tsentra (AFTs), Moscow (1999)\nKashin, B.S., Saakyan, A.A.: Orthogonal series. Aktuarno-Finans, Tsentra (AFTs), Moscow (1999)\nKokilashvili, V., Yildirir, Y.E.: Trigonometric polynomials in weighted Lorentz spaces. J. Funct. Spaces Appl. 8(1), 67–86 (2010)\nKrein, S.G., Petunin, YuI, Semenov, E.M.: Interpolation of linear operators. Nauka, Moscow (1978)\nLapin S.V.: Some embedding theorems for products of functions, Manuscript N 1036-80Dep, deposited at VINITI (Russian). pp. 31 (1980)\nLizorkin, P.I.: Generalized Holder spaces \\(B_{p, \\theta }^{(r)}\\) and their relations with the Sobolev spaces \\(L_{p}^{(r)}\\). Sib. Mat. Zhur. 9(5), 1127–1152 (1968)\nNikol’skii, S.M.: Approximation of functions of several variables and embedding theorems. Nauka, Moscow (1977)\nRomanyuk, A.S.: The approximation of the isotropic classes \\(B_{p, \\theta }^{r}\\) of periodic functions of many variables in the space \\(L_{q}\\). Tr. Inst. Mat. Ukrain. 5(1), 263–278 (2008)\nSemenov, E.M.: Interpolation of linear operators in symmetric spaces. Sov. Math. Dokl. 6(), 1294–1298 (1965)\nSharpley, R.: Space \\(\\Lambda _{\\alpha }(X)\\) and interpolation. J. Funct. Anal. 11, 479–513 (1972)\nSherstneva, L.A.: On the properties of best Lorentz approximations and certain embedding theorems. Izvestiya Vysshikh Uchebnykh Zavedenii. Matem. 10, 48–58 (1987)\nSimonov, B.V.: Embedding Nikol’skii classes into Lorentz spaces. Sib. Math. J. 51(4), 728–744 (2010)\nStasyuk, S.A.: Approximating characteristics of the analogs of Besov classes with logarithmic smoothness. Ukr. Math. J. 66(4), 553–560 (2014)\nStasyuk, S.A.: Kolmogorov widths for analogs ofthe Nikol’skii - Besov classes with logarithmic smoothness. Ukr. Math. J. 67(11), 1786–1792 (2015)\nStein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press, Princeton (1971)\nTemirgaliev, N.: On the embedding of the classes \\(H_{p}^{\\omega }\\) in Lorentz spaces. Sib. Mat. Zh. 24(2), 160–172 (1983)\nTemlyakov, V.N.: Approximation of functions with bounded mixed derivative. Tr. Mat. Inst. Steklov. 178, 3–112 (1986)\nTemlyakov, V.: Multivariate approximation. Cambridge University Press, Cambridge (2018)",{"EN":725},"In this paper, we consider the generalized Lorentz space of periodic functions of several variables and the Nikol’skii–Besov space of functions. The article establishes a sufficient condition for a function to belong from one generalized Lorentz space to another space in terms of the difference of the partial sums of the Fourier series of a given function. 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T.: Concavity of certain maps on positive definite matrices and applications to Hadamard Products. Linear Algebra Appl. 26, 203–241 (1979)",{"doi":1153},"10.1016\u002F0024-3795(79)90179-4",{"id":20,"text":1155,"url":20,"identifiers":1156},"Ando, T., Li, C.-K., Mathias, R.: Geometric means. Linear Algebra Appl. 385, 305–334 (2004)",{"doi":1157},"10.1016\u002Fj.laa.2003.11.019",{"id":20,"text":1159,"url":20,"identifiers":1160},"Bhatia, R.: Matrix Analysis. Springer-Verlag, New York (1996)",{},{"id":20,"text":1162,"url":20,"identifiers":1163},"Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)",{},{"id":20,"text":1165,"url":20,"identifiers":1166},"Bhatia, R., Holbrook, J.: Riemannian geometry and matrix geometric means. Linear Algebra Appl. 413, 594–618 (2006)",{"doi":1167},"10.1016\u002Fj.laa.2005.08.025",{"id":20,"text":1169,"url":20,"identifiers":1170},"Bhatia, R., Karandikar, R.L.: Monotonicity of the matrix geometric mean. Math. 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Linear Multilinear Algebra 67, 2253–2281 (2019)",{"doi":1203},"10.1080\u002F03081087.2018.1488938",{"id":20,"text":1205,"url":20,"identifiers":1206},"Kim, S., Lee, H.: The power mean and the least squares mean of probability measures on the space of positive definite matrices. Linear Algebra Appl. 465, 325–346 (2015)",{"doi":1207},"10.1016\u002Fj.laa.2014.09.042",{"id":20,"text":1209,"url":20,"identifiers":1210},"Kim, S., Lee, H., Lim, Y.: A fixed point mean approximation to the Cartan barycenter of positive definite matrices. Linear Algebra Appl. 496, 420–437 (2016)",{"doi":1211},"10.1016\u002Fj.laa.2016.02.005",{"id":20,"text":1213,"url":20,"identifiers":1214},"Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246, 205–224 (1980)",{"doi":1215},"10.1007\u002FBF01371042",{"id":20,"text":1217,"url":20,"identifiers":1218},"Lawson, J., Lim, Y.: Monotonic properties of the least squares mean. Math. 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Linear Algebra Appl. 463, 134–153 (2014)",{"doi":1250},"10.1016\u002Fj.laa.2014.08.025",{"id":20,"text":1252,"url":20,"identifiers":1253},"Pusz, W., Woronowicz, S.L.: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8, 159–170 (1975)",{"doi":1254},"10.1016\u002F0034-4877(75)90061-0",{"id":20,"text":1256,"url":20,"identifiers":1257},"Sturm, K.-T.: Probability measures on metric spaces of nonpositive curvature, in: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), pp. 357–390, Contemp. Math., 338, Amer. Math. Soc., Providence, RI (2003)",{"doi":1258},"10.1090\u002Fconm\u002F338\u002F06080",{"id":20,"text":1260,"url":20,"identifiers":1261},"Thompson, A.C.: On certain contraction mappings in a partially ordered vector space. Proc. Amer. Math. Soc. 14, 438–443 (1963)",{},{"id":20,"text":1263,"url":20,"identifiers":1264},"Udagawa, Y., Yamazaki, T., Yanagida, M.: Some properties of weighted operator means and characterizations of interpolational means. Linear Algebra Appl. 517, 217–234 (2017)",{"doi":1265},"10.1016\u002Fj.laa.2016.12.017",{"id":20,"text":1267,"url":20,"identifiers":1268},"Yamazaki, T.: The Riemannian mean and matrix inequalities related to the Ando-Hiai inequality and chaotic order. Oper. Matrices 6, 577–588 (2012)",{"doi":1269},"10.7153\u002Foam-06-39",{"id":20,"text":1271,"url":20,"identifiers":1272},"Yamazaki, T.: An elementary proof of arithmetic-geometric mean inequality of the weighted Riemannian mean of positive definite matrices. Linear Algebra Appl. 438, 1564–1569 (2013)",{"doi":1273},"10.1016\u002Fj.laa.2011.12.006",{"id":1275,"createTime":1276,"updateTime":1277,"relativeEntities":1278,"slug":1279,"properties":1280,"entityType":153,"verifyStatus":154,"verifyTime":1277,"verifyNote":155,"syncStatus":19,"languages":1290,"translateLanguages":20,"viewCount":21,"primaryUrl":1291,"fullTextUrl":20,"authors":1292,"publicationType":186,"publisherRelationship":1331,"citationCount":21,"citationInfo":1360,"publishDate":300,"publishYear":217,"citationAnalyzeStatus":19,"lastCitationAnalyze":20,"indexDatabases":20,"openAccess":20,"references":1362,"isForceReanalyzing":218},"fc855916-dc11-43cb-987e-5e3b364a7d7f","2024-04-14T04:36:57.277+00:00","2025-01-09T23:12:37.997+00:00",[],"Two-classes-of-operators-related-to-the-perturbation-classes-problem",{"keywords":1281,"openalex":1282,"abstract":1284,"title":1286,"doi":1288},{},{"VOID":1283},"W4376276079",{"EN":1285},"\u003Cjats:title>Abstract\u003C\u002Fjats:title>\u003Cjats:p>Let \u003Cjats:inline-formula>\u003Cjats:alternatives>\u003Cjats:tex-math>$${{\\mathcal {S}}}{{\\mathcal {S}}}$$\u003C\u002Fjats:tex-math>\u003Cmml:math xmlns:mml=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\n                  \u003Cmml:mrow>\n                    \u003Cmml:mi>S\u003C\u002Fmml:mi>\n                    \u003Cmml:mi>S\u003C\u002Fmml:mi>\n                  \u003C\u002Fmml:mrow>\n                \u003C\u002Fmml:math>\u003C\u002Fjats:alternatives>\u003C\u002Fjats:inline-formula> and \u003Cjats:inline-formula>\u003Cjats:alternatives>\u003Cjats:tex-math>$${{\\mathcal {S}}}{{\\mathcal {C}}}$$\u003C\u002Fjats:tex-math>\u003Cmml:math xmlns:mml=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\n                  \u003Cmml:mrow>\n                    \u003Cmml:mi>S\u003C\u002Fmml:mi>\n                    \u003Cmml:mi>C\u003C\u002Fmml:mi>\n                  \u003C\u002Fmml:mrow>\n                \u003C\u002Fmml:math>\u003C\u002Fjats:alternatives>\u003C\u002Fjats:inline-formula> be the strictly singular and the strictly cosingular operators acting between Banach spaces, and let \u003Cjats:inline-formula>\u003Cjats:alternatives>\u003Cjats:tex-math>$$P\\Phi _+$$\u003C\u002Fjats:tex-math>\u003Cmml:math xmlns:mml=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\n                  \u003Cmml:mrow>\n                    \u003Cmml:mi>P\u003C\u002Fmml:mi>\n                    \u003Cmml:msub>\n                      \u003Cmml:mi>Φ\u003C\u002Fmml:mi>\n                      \u003Cmml:mo>+\u003C\u002Fmml:mo>\n                    \u003C\u002Fmml:msub>\n                  \u003C\u002Fmml:mrow>\n                \u003C\u002Fmml:math>\u003C\u002Fjats:alternatives>\u003C\u002Fjats:inline-formula> and \u003Cjats:inline-formula>\u003Cjats:alternatives>\u003Cjats:tex-math>$$P\\Phi _+$$\u003C\u002Fjats:tex-math>\u003Cmml:math xmlns:mml=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\n                  \u003Cmml:mrow>\n                    \u003Cmml:mi>P\u003C\u002Fmml:mi>\n                    \u003Cmml:msub>\n                      \u003Cmml:mi>Φ\u003C\u002Fmml:mi>\n                      \u003Cmml:mo>+\u003C\u002Fmml:mo>\n                    \u003C\u002Fmml:msub>\n                  \u003C\u002Fmml:mrow>\n                \u003C\u002Fmml:math>\u003C\u002Fjats:alternatives>\u003C\u002Fjats:inline-formula> be the perturbation classes for the upper and the lower semi-Fredholm operators. We study two classes of operators \u003Cjats:inline-formula>\u003Cjats:alternatives>\u003Cjats:tex-math>$$\\Phi {\\mathcal {S}}$$\u003C\u002Fjats:tex-math>\u003Cmml:math xmlns:mml=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\n                  \u003Cmml:mrow>\n                    \u003Cmml:mi>Φ\u003C\u002Fmml:mi>\n                    \u003Cmml:mi>S\u003C\u002Fmml:mi>\n                  \u003C\u002Fmml:mrow>\n                \u003C\u002Fmml:math>\u003C\u002Fjats:alternatives>\u003C\u002Fjats:inline-formula> and \u003Cjats:inline-formula>\u003Cjats:alternatives>\u003Cjats:tex-math>$$\\Phi {\\mathcal {C}}$$\u003C\u002Fjats:tex-math>\u003Cmml:math xmlns:mml=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\n                  \u003Cmml:mrow>\n                    \u003Cmml:mi>Φ\u003C\u002Fmml:mi>\n                    \u003Cmml:mi>C\u003C\u002Fmml:mi>\n                  \u003C\u002Fmml:mrow>\n                \u003C\u002Fmml:math>\u003C\u002Fjats:alternatives>\u003C\u002Fjats:inline-formula> that satisfy \u003Cjats:inline-formula>\u003Cjats:alternatives>\u003Cjats:tex-math>$${{\\mathcal {S}}}{{\\mathcal {S}}}\\subset \\Phi {\\mathcal {S}}\\subset P\\Phi _+$$\u003C\u002Fjats:tex-math>\u003Cmml:math xmlns:mml=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\n                  \u003Cmml:mrow>\n                    \u003Cmml:mi>S\u003C\u002Fmml:mi>\n                    \u003Cmml:mi>S\u003C\u002Fmml:mi>\n                    \u003Cmml:mo>⊂\u003C\u002Fmml:mo>\n                    \u003Cmml:mi>Φ\u003C\u002Fmml:mi>\n                    \u003Cmml:mi>S\u003C\u002Fmml:mi>\n                    \u003Cmml:mo>⊂\u003C\u002Fmml:mo>\n                    \u003Cmml:mi>P\u003C\u002Fmml:mi>\n                    \u003Cmml:msub>\n                      \u003Cmml:mi>Φ\u003C\u002Fmml:mi>\n                      \u003Cmml:mo>+\u003C\u002Fmml:mo>\n                    \u003C\u002Fmml:msub>\n                  \u003C\u002Fmml:mrow>\n                \u003C\u002Fmml:math>\u003C\u002Fjats:alternatives>\u003C\u002Fjats:inline-formula> and \u003Cjats:inline-formula>\u003Cjats:alternatives>\u003Cjats:tex-math>$${{\\mathcal {S}}}{{\\mathcal {C}}}\\subset \\Phi {\\mathcal {C}}\\subset P\\Phi _-.$$\u003C\u002Fjats:tex-math>\u003Cmml:math xmlns:mml=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\n                  \u003Cmml:mrow>\n                    \u003Cmml:mi>S\u003C\u002Fmml:mi>\n                    \u003Cmml:mi>C\u003C\u002Fmml:mi>\n                    \u003Cmml:mo>⊂\u003C\u002Fmml:mo>\n                    \u003Cmml:mi>Φ\u003C\u002Fmml:mi>\n                    \u003Cmml:mi>C\u003C\u002Fmml:mi>\n                    \u003Cmml:mo>⊂\u003C\u002Fmml:mo>\n                    \u003Cmml:mi>P\u003C\u002Fmml:mi>\n                    \u003Cmml:msub>\n                      \u003Cmml:mi>Φ\u003C\u002Fmml:mi>\n                      \u003Cmml:mo>-\u003C\u002Fmml:mo>\n                    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Stud. Math. 131, 271–287 (1998)",{},{"id":20,"text":1367,"url":20,"identifiers":1368},"Aiena, P., González, M., Martínez-Abejón, A.: Characterizations of strictly singular and strictly cosingular operators by perturbation classes. Glasg. Math. J. 54, 87–96 (2011)",{"doi":1369},"10.1017\u002FS0017089511000346",{"id":20,"text":1371,"url":20,"identifiers":1372},"Argyros, S.A., Felouzis, V.: Interpolating hereditarily indecomposable Banach spaces. J. Am. Math. Soc. 13, 243–294 (2000)",{"doi":1373},"10.1090\u002FS0894-0347-00-00325-8",{"id":20,"text":1375,"url":20,"identifiers":1376},"Bessaga, C., Pełczyński, A.: Banach spaces non-isomorphic to their Cartesian squares. I. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys. 8, 77–80 (1960)",{},{"id":20,"text":1378,"url":20,"identifiers":1379},"Caradus, S., Pfaffenberger, W., Yood, B.: Calkin Algebras and Algebras of Operators in Banach Spaces. Lecture Notes in Pure and Applies Mathematics, M. Dekker, New York (1974)",{},{"id":20,"text":1381,"url":20,"identifiers":1382},"Friedman, T.L.: Relating strictly singular operators to the condition $$X \u003C Y \\; mod \\;({\\cal{S} }, {\\cal{T} })$$ and resulting perturbations. Analysis (Munich) 22, 347–354 (2002)",{},{"id":20,"text":1384,"url":20,"identifiers":1385},"Galego, E.M., González, M., Pello, J.: On subprojectivity and superprojectivity. Results Math. 71, 1191–1205 (2017)",{"doi":1386},"10.1007\u002Fs00025-016-0558-3",{"id":20,"text":1388,"url":20,"identifiers":1389},"Giménez, J., González, M., Martínez-Abejón, A.: Perturbation of semi-Fredholm operators on products of Banach spaces. J. Oper. Theory 68, 501–514 (2012)",{},{"id":20,"text":1391,"url":20,"identifiers":1392},"Gohberg, I.C., Markus, A.S., Feldman, I.A.: Normally solvable operators and ideals associated with them. Bul. Akad. Štiince RSS Moldoven 10(76), 51–70 (1960). [Translation: Am. Math. Soc. Transl. (2) 61, 63–84 (1967)]",{},{"id":20,"text":1394,"url":20,"identifiers":1395},"González, M.: The perturbation classes problem in Fredholm theory. J. Funct. Anal. 200, 65–70 (2003)",{"doi":1396},"10.1016\u002FS0022-1236(02)00071-X",{"id":20,"text":1398,"url":20,"identifiers":1399},"González, M., Martínez-Abejón, A., Salas-Brown, M.: Perturbation classes for semi-Fredholm operators on subprojective and superprojective spaces. Ann. Acad. Sci. Fenn. Math. 36, 481–491 (2011)",{"doi":1400},"10.5186\u002Faasfm.2011.3625",{"id":20,"text":1402,"url":20,"identifiers":1403},"González, M., Pello, J.: Superprojective Banach spaces. J. Math. Anal. Appl. 437, 1140–1151 (2016)",{"doi":1404},"10.1016\u002Fj.jmaa.2016.01.033",{"id":20,"text":1406,"url":20,"identifiers":1407},"González, M., Pello, J., Salas-Brown, M.: Perturbation classes of semi-Fredholm operators in Banach lattices. J. Math. Anal. Appl. 420, 792–800 (2014)",{"doi":1408},"10.1016\u002Fj.jmaa.2014.06.012",{"id":20,"text":1410,"url":20,"identifiers":1411},"González, M., Pello, J., Salas-Brown, M.: The perturbation classes problem for subprojective and superprojective Banach spaces. J. Math. Anal. Appl. 489, 124191 (2020)",{"doi":1412},"10.1016\u002Fj.jmaa.2020.124191",{"id":20,"text":1414,"url":20,"identifiers":1415},"González, M., Salas-Brown, M.: Perturbation classes for semi-Fredholm operators in $$L_p(\\mu )$$-spaces. J. Math. Anal. Appl. 370, 11–17 (2010)",{"doi":1416},"10.1016\u002Fj.jmaa.2010.04.051",{"id":20,"text":1418,"url":20,"identifiers":1419},"Gowers, W.T., Maurey, B.: The unconditional basic sequence problem. J. Am. Math. Soc. 6, 851–874 (1993)",{"doi":1420},"10.1090\u002FS0894-0347-1993-1201238-0",{"id":20,"text":1422,"url":20,"identifiers":1423},"Gowers, W.T., Maurey, B.: Banach spaces with small spaces of operators. Math. Ann. 307, 543–568 (1997)",{"doi":1424},"10.1007\u002Fs002080050050",{"id":20,"text":1426,"url":20,"identifiers":1427},"Oikhberg, T., Spinu, E.: Subprojective Banach spaces. J. Math. Anal. Appl. 424, 613–635 (2015)",{"doi":1428},"10.1016\u002Fj.jmaa.2014.11.008",{"id":20,"text":1430,"url":20,"identifiers":1431},"Pietsch, A.: Operator Ideals. North-Holland, Amsterdam (1980)",{},{"id":20,"text":1433,"url":20,"identifiers":1434},"Semadeni, Z.: Banach spaces non-isomorphic to their Cartesian squares. II. Bull. Acad. Pol. Sci. Sér. Sci. Math. Astron. Phys. 8, 81–84 (1960)",{},{"id":20,"text":1436,"url":20,"identifiers":1437},"Weis, L.: Perturbation classes of semi-Fredholm operators. Math. Z. 178, 429–442 (1981)",{"doi":1438},"10.1007\u002FBF01214880",{"id":20,"text":1440,"url":20,"identifiers":1441},"Whitley, R.J.: Strictly singular operators and their conjugates. Trans. Am. Math. Soc. 113, 252–261 (1964)",{"doi":1442},"10.1090\u002FS0002-9947-1964-0177302-2"]