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On the mathematical work of Mikhaĭl Zakharovich Solomyak. (English. Russian original) Zbl 1436.47001

St. Petersbg. Math. J. 30, No. 3, 391-410 (2019); translation from Algebra Anal. 30, No. 3, 3-29 (2018).
This is an epitaph and tribute to {М. Z. Solomyak} (1931–2016) who contributed greatly to functional analysis and spectral theory. Alongside a short biography, the article surveys {M. Z. Solomyak}’s studies on differential equations in Banach spaces and the theory of elliptic systems as well as on his joint research with {M. Sh. Birman} concerning mainly estimates and asymptotics of the eigenvalues and singular values of integral operators, asymptotics of the spectra of differential operators, nonstandard spectral problems, etc. The complete list of {М. Z. Solomyak}’s publications is attached.

MSC:

47-03 History of operator theory
35-03 History of partial differential equations
01A70 Biographies, obituaries, personalia, bibliographies
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35P15 Estimates of eigenvalues in context of PDEs
47F05 General theory of partial differential operators
47B25 Linear symmetric and selfadjoint operators (unbounded)

Biographic References:

Solomyak, Mikhaĭl Zakharovich
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References:

[1] Sol1953 M. Z. Solomyak, On characteristic values and characteristic vectors of a perturbed operator, Dokl. Akad. Nauk SSSR 90 (1953), no. 1, 29-32. (Russian)
[2] Sol1957 M. Z. Solomyak, On orthogonal based in Banach space, Vestnik Leningrad. Univ. 12 (1957), no. 1, 27-36. (Russian) · Zbl 0093.12102
[3] Sol1958 M. Z. Solomyak, Application of semigroup theory to the study of differential equations in Banach spaces, Dokl. Akad. Nauk SSSR 122 (1958), no. 5, 766-769. (Russian) · Zbl 0090.09902
[4] Sol1959 M. Z. Solomyak, Analyticity of a semigroup generated by an elliptic operator in \(L_p\) spases, Dokl. Akad. Nauk SSSR 127 (1959), no. 1, 37-39. (Russian) · Zbl 0087.11703
[5] Sol1960_IzvV M. Z. Solomyak, On differential equations in Banach spaces, Izv. Vyssh. Uchebn. Zaved. Mat. 1960, no. 1, 198-209. (Russian) · Zbl 0097.10403
[6] Sol1960_UMN M. Z. Solomyak, Evalution of norm of the resolvent of elliptic operators in \(L_p\), Uspekhi Mat. Nauk 15 (1960), no. 6, 141-148. (Russian)
[7] Sol1961 M. Z. Solomyak, Elliptic operators on two-dimensional manifolds, Dokl. Akad. Nauk SSSR 139 (1961), no. 1, 37-39; English transl., Soviet Math. Dokl. 2 (1961), 870-872. · Zbl 0112.38501
[8] Sol1962_DAN M. Z. Solomyak, On the spaces conjugate to the \(W_p^l\) spaces of S. L. Sobolev, Dokl. Akad. Nauk SSSR 143 (1962), no. 6, 1289-1292; English transl., Soviet Math. Dokl. 3 (1962), 605-608. · Zbl 0121.09602
[9] Sol1962_Gertzen M. Z. Solomyak, On the solvability of certain integral identities, Leningrad. Gos. Ped. Inst. Uchen. Zap. 238 (1962), 141-148. (Russian)
[10] Sol1963_DAN M. Z. Solomyak, On linear elliptic systems of first order, Dokl. Akad. Nauk SSSR 150 (1963), no. 1, 48-51; English transl., Soviet Math. Dokl. 4 (1963), 604-607. · Zbl 0163.34602
[11] 11 M. Z. Solomyak, On local properties of factional powers if elliptic operators, Uspekhi Mat. Nauk 18 (1963), no. 2, 184-185. (Russian)
[12] BSol1965 M. Sh. Birman and M. Z. Solomyak, On double Stieltjes operator integrals, Dokl. Akad. Nauk SSSR 165 (1965), no. 6, 1223-1226; English transl., Soviet Math. Dokl. 6 (1965), no. 2, 1567-1571. · Zbl 0143.36803
[13] new-1 M. Z. Solomyak, On Lopatinski\i condition for the solvability of boundary problems, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1965 vyp. 1, 143-149. (Russian)
[14] BSol1966 M. Sh. Birman and M. Z. Solomyak, Approximation of functions of the \(W_p^\)-classes by piece-wise-polynomial functions, Dokl. Akad. Nauk SSSR 171 (1966), no. 5, 1015-1018; English transl., Soviet Math. Dokl. 7 (1966), no. 2, 1573-1577. · Zbl 0173.15903
[15] BSol1966_DAN M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals and problems on multipliers, Dokl. Akad. Nauk SSSR 171 (1966), no. 6, 1251-1254; English transl., Soviet Math. Dokl. 7 (1966), no. 2, 1618-1621. · Zbl 0177.18801
[16] BSol1966_LGU M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals, Spectral Theory and Wave Processes, Probl. Mat. Fiz., vyp. 1, Leningrad. Univ., Leningrad, 1966, pp. 33-67; English transl., Topics Math. Phys., vol. 1, Consultants Bureau Plenum Publ. Co., New York, 1967, pp. 25-54. · Zbl 0161.34602
[17] BSol1967_Vestn2 M. Sh. Birman and M. Z. Solomyak, Estimates of singular numbers of integral operators. I, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1967, vyp. 2, 43-53. (Russian) · Zbl 0161.34701
[18] BSol1967_MatSb M. Sh. Birman and M. Z. Solomyak, Piecewise polynomial approximations of functions of classes \(W_p^\), Mat. Sb. 73 (1967), no. 3, 331-355; English transl., Math. USSR – Sb. 2 (1967), 295-317. · Zbl 0173.16001
[19] BSol1967_Vestn3 M. Sh. Birman and M. Z. Solomyak, Estimates of singular numbers of integral opeators. II, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1967, vyp. 3, 21-28. (Russian) · Zbl 0177.18802
[20] SolTi M. Z. Solomyak and V. M. Tikhomirov, Geometric characteristics of the embedding of the classes \(W_p^\) in \(C\), Izv. Vyssh. Uchebn. Zaved. Mat. 1967, no. 10, 76-82. (Russian)
[21] BSol1967_LOMI M. Z. Solomyak, Transformers generated by Stieltjes double operator integrals, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 5 (1967), 201-231. (Russian)
[22] BSol_LGU_DOI2 M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals. II, Spectral Theory, Diffraction Problems, Probl. Mat. Fiz., vyp. 2, Leningrad. Univ., Leningrad, 1967, pp. 26-60; English transl., Topics Math. Phys., vol. 2, Consultants Bureau, New York, 1968, pp. 19-46. · Zbl 0182.46202
[23] 22 M. Z. Solomyak, The approximation of double Stieltjes operator integrals by integral sums, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1968, vyp. 7, 148-149. (Russian) · Zbl 0165.48903
[24] BSol1968_LGU M. Z. Solomyak, Double operator integrals in the ring \(\hat R\), Probl. Mat. Fiz., vyp. 3, Leningrad. Univ., Leningrad, 1968, pp. 88-102; English transl., Topics Math. Phys. 3 (1969), 79-91.
[25] BSol1969_Vestn1 M. Sh. Birman and M. Z. Solomyak, Estimates of singular numbers of integral operators. III, Operators domains, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1969, vyp. 1, 35-48. (Russian) · Zbl 0181.13601
[26] 25 M. Sh. Birman and M. Z. Solomyak, Remarks on the nuclearity of integral operators and the boundedness of pseudodifferential operators, Izv. Vyssh. Uchebn. Zaved. Mat. 1969, no. 9, 11-17. (Russian) · Zbl 0187.08501
[27] SolSt1969 M. Z. Solomyak and V. V. Sten\textprime kin, A certain class of multiple operator Stieltjes integrals, Probl. Mat. Anal., vyp. 2, Leningrad. Univ., 1969, pp. 122-134. (Russian)
[28] 27 M. Z. Solomyak, Estimates of singular numbers of integral operators. IV, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1970, vyp. 1, 76-87. (Russian) · Zbl 0204.45602
[29] BSol1970_FAA M. Sh. Birman and M. Z. Solomyak, The principal term of spectral asymptotics for “nonsmooth” elliptic problems, Funktional. Anal. i Prilozhen. 4 (1970), no. 4, 1-13; English transl., Funct. Anal. Appl. 4 (1970), 265-275. · Zbl 0225.35077
[30] BSol1970_DAN M. Sh. Birman and M. Z. Solomyak, Asymptotics of the spectrum of weakly polar integraal operators, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), no. 5, 1142-1158; English transl., Math. USSR – Izv. 4 (1970), 1151-1168. · Zbl 0261.47027
[31] 30 M. Sh. Birman and M. Z. Solomyak, On the proof of the theorem on the asymptotic behavior of the spectrum of the real part of a Volterra operator with nuclear imaginary component, Math. Issled. 5 (1970), no. 4, 16-25. (Russian) · Zbl 0224.47010
[32] BSol1971_FAA M. Sh. Birman and M. Z. Solomyak, The asymptotics of the spectrum of “nonsmooth” elliptic equations, Funktsional. Anal. i Prilozhen. 5 (1971), no. 1, 69-70; English transl., Funct. Anal. Appl. 5 (1971), 56-57. · Zbl 0226.35072
[33] 32 G. P. Kostometov and M. Z. Solomyak, Estimates of the singular values of integral operators with a weak singularity, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1971, vyp. 1, 28-39. (Russian) · Zbl 0214.12303
[34] 33 A. A. Kruglov and M. Z. Solomyak, Interpolation of operators in the spaces \(V_p\), Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1971, vyp. 13, 54-60. (Russian) · Zbl 0224.46039
[35] 34 V. M. Babich, M. Sh. Birman, V. I. Smirnov, M. Z. Solomyak, and N. N. Ural\textprime tseva, Ol \textprime ga Aleksandrovna Ladyzhenskaya (on her fiftieth birthday), Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1972, vyp. 7, 159-160. (Russian)
[36] BSol1972_DAN M. Sh. Birman and M. Z. Solomyak, Spectral asymptotic behavior of nonsmooth elliptic operators, Dokl. Akad. Nauk SSSR 205 (1972), no. 2, 267-270; English transl., Soviet Math. Dokl. 13 (1972), 906-910. · Zbl 0264.35056
[37] VSol1972 I. L. Vules and M. Z. Solomyak, Spectral asymptotic behavior of degenerate elliptic operators, Dokl. Akad. Nauk SSSR 207 (1972), no. 2, 262-265; English transl., Soviet Math. Dokl. 13 (1972), 1484-1488. · Zbl 0268.35070
[38] BSol1972_LOMI M. Sh. Birman and M. Z. Solomyak, Remarks on the spectral shift function, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 27 (1972), 33-46; English transl., J. Soviet Math. 3 (1975), no. 4, 408-419. · Zbl 0329.47009
[39] BSol1972_TMMO M. Sh. Birman and M. Z. Solomyak, Spectral asymptotics of nonsmooth elliptic operators. I, II, Tr. Moskov. Mat. Obshch. 27 (1972), 3-52; 28 (1973), 3-34; English transl., Trans. Moscow Math. Soc. 27 (1972), 1-52; 28 (1973), 1-32. · Zbl 0251.35075
[40] 39 V. I. Matsaev and M. Z. Solomyak, On existence conditions for the Stieltjes integral, Mat. Sb. 88 (1972), no. 4, 522-535; English transl., Math. USSR-Sb. 17 (1972), no. 4, 515-527. · Zbl 0255.26007
[41] VSol1973 I. L. Vules and M. Z. Solomyak, Spectral asymptotic analysis of the degenerate Steklov problem, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1973, vyp. 4, 148-150. (Russian)
[42] BSol1973_IzvV M. Sh. Birman and M. Z. Solomyak, Equivalent renormings of \(W^r_p\) classes in the case of partitioning of the domain, Izv. Vyssh. Uchebn. Zaved. 1973, no. 3, 19-27. (Russian) · Zbl 0269.46024
[43] BSol1973_LGU M. Sh. Birman and M. Z. Solomyak, Double Stieltjes operator integrals. III, Probl. Mat. Fiz., vyp. 6, Leningrad. Univ., Leningrad, 1973, pp. 27-53. (Russian)
[44] VSol1974 I. L. Vules and M. Z. Solomyak, Spectral asymptotics of second-order degenerate elliptic operators, Izv. Akad. Nauk SSR Ser. Mat. 38 (1974), no. 6, 1362-1392; English transl., Math. USSR-Izv. 8 (1974), no. 6, 1343-1371. · Zbl 0367.35042
[45] BSol_10School M. Sh. Birman and M. Z. Solomyak, Quantitative analysis in Sobolev’s imbedding theorems and applications to spectral theory, Tenth Math. School (Summer School, Kaciveli/Nalchik, 1972), Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1974, pp. 5-189. (Russian)
[46] 45 M. Sh. Birman and M. Z. Solomyak, A certain “model” nonelliptic spectral problem, Collection of Articles Dedicated to the Memory of Academician V.I.Smirnov, Vestnik Leningrad Univ. Ser. Mat. Mekh. Astronom. 1975, vyp. 1, 39-45; English transl., Vestnik Leningrad. Univ. Math. 8 (1980), 23-30.
[47] 46 M. Sh. Birman, L. S. Koplipenko, and M. Z. Solomyak, Estimates of the spectrum of a difference of fractional powers of self-adjoint operators, Izv. Vyssh. Uchebn. Zaved. Mat. 1975, no. 3, 3-10; English transl., Soviet Math. (Iz. VUZ) 19 (1975), no. 3, 1-6.
[48] BSol1977_UMN M. Sh. Birman and M. Z. Solomyak, Estimates for the singular numbers of integral operators, Uspekhi Mat. Nauk 32 (1977), no. 1, 17-84; English transl., Russian Math. Surveys 32 (1977), no. 1, 15-89. · Zbl 0376.47023
[49] BSol_vestnik1977-3 M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of pseudodifferentials operators with anisotropically homogeneous symbols, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1977, vyp. 3, 13-21; English transl., Vestnik Leningrad. Univ. Math. 10, (1982), 237-247. · Zbl 0377.47033
[50] BSol_VINITI_1977 M. Sh. Birman and M. Z. Solomyak, Asymptotic properties of the spectrum of differential equation, Itogi Nauki i Tehniki Mat. Anal., vol. 14, VINITI, Moscow, 1977, pp. 5-58; English transl., J. Soviet Math. 12, (1979), no. 3, 247-283. · Zbl 0424.35069
[51] BSol1977_Sibir M. Sh. Birman and M. Z. Solomyak, Application of interpolational methods to estimates of the spectrum of integral operators, Operator Theory in Function Spaces (Proc. School, Novosibirsk, 1975), Nauka. Novosibirsk, 1977, pp. 42-70; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 127, Amer. Math. Soc., Providence, Rl, 1985, pp. 61-82.
[52] 51 M. Sh. Birman and M. Z. Solomyak, On asymptotic behavior of the spectrum of elliptic variational problems with constraints, Proc. All-Union Conf. on Partial Differential Equations Dedicated to Acad. Petrovskiy 75th Birthday, Moskov. Univ., Moscow, 1978, pp. 51-52. (Russian) · Zbl 0498.35064
[53] BSol1979_SibJ M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations, Sibirsk. Mat. Zh. 20 (1979), no. 1, 3-22; English transl., Sib. Math. J. 20 (1979), no. 1, 1-15. · Zbl 0427.35050
[54] 53 M. Sh. Birman, A. M. Vershik, and M. Z. Solomyak, The product of commuting spectral measures may fail to be countably additive, Funktsional. Anal. i Prilozhen. 13 (1979), no. 1, 61-62; English transl., Funct. Anal. Appl. 13 (1979), no. 1, 48-49. · Zbl 0423.28008
[55] BSol1979 M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of pseudodifferential operators with anisotropically homogeneous sumbols, II, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1979, vyp. 3, 5-10; English transl., Vestnik Leningrad. Univ. Math. 12, (1980), 155-161.
[56] BSol_10School_tr M. Sh. Birman and M. Z. Solomyak, Quantitative analysis in Sobolev imbedding theorems and applications to spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 114, Amer. Math. Soc., Providence, RI, 1980. · Zbl 0426.46020
[57] 56 S. Ulare and M. Z. Solomyak, Spaces dual to the spaces \(\Lambda_\), Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1980, vyp. 1, 113-115; English transl., Vestnik Leningrad. Univ. Math. · Zbl 0438.46015
[58] BSol1980 M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations in unbounded domains, Funktsional. Anal. Prilozhen. 14 (1980), no. 4, 27-35; English transl., Funct. Anal. Appl. 14 (1980), 267-274. · Zbl 0463.35061
[59] BSol1980_Trudy M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations, Partial Differential Equations (Proc. Conf. Novosibirsk, 1978), Nauka, Novosibirsk, 1980, pp. 221-223.
[60] BSol_book1980 M. Sh. Birman and M. Z. Solomyak, Spectral theory of self-adjoint operators in Hilbert space, Leningrad. Univ., Leningrad, 1980. (Russian)
[61] 61 N. A. Karazeeeva and M. Z. Solomyak, The asymptotic behavior of the spectrum of a contact problem for second-order elliptic equations, Probl. Mat. Anal., vyp. 8, Leningrad. Univ., Leningrad, 1981, pp. 36-48. · Zbl 0484.35065
[62] BSol1982_Vestnik1 M. Sh. Birman and M. Z. Solomyak, On subspaces that admit a differential projector, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1982, vyp. 1, 18-25; English transl., Vestnik Leningrad. Univ. Math. 15 (1983), 17-27. · Zbl 0484.35085
[63] BSol_LOMI1982 M. Sh. Birman and M. Z. Solomyak, The asymptotic behavior of the spectrum of variational problems on solutions of elliptic systems, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982), 23-39; English transl., J. Soviet Math. 28 (1985), 633-644. · Zbl 0501.35065
[64] 64 A. I. Karol\textprime and M. Z. Solomyak, Class of differential operators with abnormal behavior of the spectrum of the Dirichlet problem, Mat. Zametki 31 (1982), no. 5, 747-751; English transl., Math. Notes 31 (1982), no. 5, 379-381. · Zbl 0515.35066
[65] 65 M. Sh. Birman and M. Z. Solomyak, Asymptotic behavior of the spectrum of pseudodifferential variational problems with shifts, Probl. Mat. Fiz., no. 10, Leningrad. Univ., Leningrad, 1982, pp. 20-36; English transl., Selecta Math. Sov. 5 (1986), 245-246.
[66] BSol1983_LOMI M. Sh. Birman and M. Z. Solomyak, Contact operators with power asymptotic behavior of the singular number, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 126 (1983), 21-30; English transl., J. Soviet Math. 27 (1984), 2442-2447. · Zbl 0518.47014
[67] Sol1983 M. Z. Solomyak, The Calderon projection, Operator Theory and Function Theory, no. 1, Leningrad. Univ., Leningrad, 1983, pp. 47-55. (Russian)
[68] Sol1984 M. Z. Solomyak, Asymptotic behavior of the spectrum of a Schr\"odinger operator with nonregular homogeneous potential, Dokl. Akad. Nauk SSSR 278 (1984), no. 2, 291-295; English transl., Soviet Math. Dokl. 30 (1984), 379-383. · Zbl 0588.35069
[69] BSol1984 M. Sh. Birman and M. Z. Solomyak, Some problems on compact operators with power-like behavior of singular numbers, Linear and Complex Analysis Problem Book. 199 Research Problems., Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, 1984, pp. 217-218.
[70] Sol1985 M. Z. Solomyak, Asymptotics of the spectrum of the Schr\"odinger operator with nonregular homogeneous potential, Mat. Sb. 127 (1985), no. 1, 21-39; English transl., Math. USSR-Sb. 55 (1986), 19-37. · Zbl 0657.35099
[71] BSol1987_SMJ M. Sh. Birman and M. Z. Solomyak, The Maxwell operator for domains with a nonsmooth boundary, Sibirsk Mat. Zh. 28 (1987), no. 1, 23-36; English transl., Sib. Math. J. 28 (1987), 12-24. · Zbl 0655.35067
[72] BSol1987_Vestnik3 M. Sh. Birman and M. Z. Solomyak, Weyl asymptotics of the spectrum of the Maxwell operator for domains with a Lipschitz boundary, Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1987, vyp. 3, 23-28; English transl., Vestnik Leningrad. Univ. Math. 20 (1987), no. 3, 15-21. · Zbl 0639.35062
[73] BSol1987_UMN M. Sh. Birman and M. Z. Solomyak, \(L_2\)-theory of the Maxwell operator in arbitary domains, Uspekhi Mat. Nauk 42 (1987), no. 6, 61-76; English transl., Russian Math. Surveys 42 (1987), no. 6, 75-96. · Zbl 0653.35075
[74] 81 M. Sh. Birman and M. Z. Solomyak, Construction in a piecewise-smooth domain of a function of the class \(H^2\) from the value of the conormal derivative, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 163 (1987), 17-28; English transl., J. Soviet Math. 49 (1990), no. 5, 1128-1136. · Zbl 0677.46042
[75] SBol_book_Engl M. Sh. Birman and M. Z. Solomyak, Spectral theory of selfadjoint operators in Hilbert space, Math. Appl. (Soviet Ser.), D. Reidel Publishing Co., Dordrecht, 1987.
[76] 75 V. S. Buslaev, I. M. Gel\textprime fand, O. A. Ladyzgenskaya, B. S. Pavlov, M. Z. Solomyak, L. D. Fadeev, and D. R. Yafaev, Mikgail Shl\"emovich Birman (on the occasion of his sixtieth birthday), Uspekhi Mat. Nauk 43 (1988), no. 3, 201-202; English transl., Russian Math. Surveys 43 (1988), no. 3, 233-235.
[77] 76 Yu. K. Demyanovich, V. P. Il\textprime in, A. I. Koshelev, M. Z. Solomyak, and N. N. Ural\textprime tseva, Solomon Grigor\textprime evich Mikhlin (on the ocassion of his 80th birthday), Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1988, vyp. 2, 128-129. (Russian)
[78] 77 S. A. Vinogradov, D. A. Vladimirov, M. K. Gavurin, S. M. Ermakov, B. M. Makarov, G. I. Natanson, M. Z. Solomyak, D. K. Fadeev, and V. P. Khavin, Grigori\vMikhailovich Fikhtengol’ts (on the \rm 100th anniversary of his birth.) Vestnik Leningrad. Univ. Ser. Mat. Mekh. Astronom. 1988, vyp. 3, 3-6. (Russian)
[79] RSolSh G. V. Rozenblum, M. Z. Solomyak, and M. A. Shubin, Spectral theory of differential operators, Itogi Nauku i Tekhniki. Sovrem. Probl. Mat. Fundam. Napravl., vol. 64 VINITI, Moscow, 1989, pp. 5-248. (Russian)
[80] BSol_Dubna_89 M. Sh. Birman and M. Z. Solomyak, Interpolation estimates for the number of negative eigenvalues of a Schr\`“odinger operator, Schr\'”odinger Operators, Standard and Nonstandard (Dubna, 1988), World Sci. Publ., Teaneck, NJ, 1989, pp.  2-18.
[81] BSol1988 M. Sh. Birman and M. Z. Solomyak, The \(L_2\)-theory of the Maxwell operator in domains with nonsmooth boundary, Symposium “Partial Differential Equations” (Holzhau, 1988), Teubner-Texte Math., vol. 112, Teubner, Leipzig, 1989, pp. 33-41.
[82] BSol1989_AA M. Sh. Birman and M. Z. Solomyak, The selfadjoint Maxwell operator in arbitrary domains, Algebra i Analiz 1 (1989), no. 1, 96-110; English transl., Leningrad Math. J. 1 (1990), no. 1, 99-115. · Zbl 0733.35099
[83] 91 M. Sh. Birman and M. Z. Solomyak, Operator integration, perturbations and commutators, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI) 170 (1989), 34-66; English transl.,\linebreak J. Soviet Math. 63 (1993), no. 2, 129-148. · Zbl 0709.47022
[84] 89 M. Sh. Birman and M. Z. Solomyak, Estimates for the difference of fractional powers of selfadjoint operators under unbounded perturbations, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989), 120-145; English transl., J. Soviet Math. 61 (1992), no. 2, 2018-2035. · Zbl 0709.47004
[85] BSol_Dubna_90 M. Sh. Birman and M. Z. Solomyak, Negative discrete spectrum of the Schr\`“odinger operator with large coupling constant: a qualitative discussion, Order, Disorder and Chaos in Quantum Systems (Dubna, 1989), Oper. Theory Adv. Appl., vol. 46, Birkh\'”auser, Basel, 1990, pp. 3-16.
[86] 84 A. D. Aleksandrov, S. S. Kutateladze, Yu. G. Reshetnyak, and M. Z. Solomyak, Viktor Abramovich Zalgaller (on the occasion of his seventieth birthday), Uspekhi Mat. Nauk 46 (1991), no. 1, 215-216; English transl., Russian Math. Surveys 46 (1991), no. 1, 257-259. · Zbl 0744.01010
[87] BSol_Varna91 M. Sh. Birman and M. Z. Solomyak, The estimates for the number of negative bound states of the Schr\"odinger operator for large coupling constants, Integral Equations and Inverse Problems (Varna, 1989), Pitman Res. Notes Math. Ser., vol. 235, Longman Sci. Tech., Harlow, 1991, pp. 49-57.
[88] 86 M. Sh. Birman and M. Z. Solomyak, Discrete negative spectrum under nonregular perturbations (polyharmonic operators, Schr\"odinger operators with a magnetic field, periodic operators, Rigorous Results in Quantum Dynamics (Liblice, 1990), World Sci. Publ., River Edge, NJ, 1991, pp. 25-36.
[89] BSol_AMS_91 M. Sh. Birman and M. Z. Solomyak, Estimates for the number of negative eigenvalues of the Schr\"odinger operator and its generalizations, Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989-90), Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 1-55. · Zbl 0749.35026
[90] BKSol M. Sh. Birman, G. E. Karadzhov, and M. Z. Solomyak, Boundedness conditions and spectrum estimates for the operators \(b(X)a(D)\) and their analogs, Estimates and Asymptotics for Discrete Spectra of Integral and Differential Equations (Leningrad, 1989-90), Adv. Soviet Math., vol. 7, Amer. Math. Soc., Providence, RI, 1991, pp. 85-106. · Zbl 0768.47025
[91] BSol1969_Novg M. Sh. Birman and M. Z. Solomyak, Schr\"odinger operator. Estimates for number of bound states as function-theoretical problem, Spectral Theory of Operators (Novgorod, 1989), Amer. Math. Soc. Transl. Ser. 2, vol. 150, Soviet Regional Conf., Amer. Math. Soc., Providence, RI, 1992, pp. 1-54.
[92] BSol1993_AA M. Sh. Birman and M. Z. Solomyak, Principal singularities of the electric component of an electromagnetic field in regions with screens, Algebra i Analiz 5 (1993), no. 1, 143-159; English transl., St. Petersburg Math. J. 5 (1994), no. 1, 125-139. · Zbl 0804.35127
[93] Sol_IEOT1994 M. Z. Solomyak, A remark on the Hardy inequalities, Integral Equations Operator Theory 19 (1994), no. 1, 120-124. · Zbl 0812.26007
[94] Sol_IJM1994 M. Z. Solomyak, Piecewise-polynomial approximation of functions from \(H^l((0,1)^d), 2l=d\), and applications to the spectral theory of the Schr\"odinger operator, Israel J. Math. 86 (1994), no. 1-3, 253-275. · Zbl 0803.47045
[95] 95 V. Olevski\i and M. Z. Solomyak, An estimate for Schur multipliers in \(S_p\)-classes, Linear Algebra Appl. 208/209 (1994), 57-64. · Zbl 0806.15020
[96] 96 M. Z. Solomyak, Eigenvalue estimates for a class of operators related to self-similar measures, Journ\'ees, \'Equations aux D\'eriv\'ees Partielles” (Saint-Jean-de-Monts, 1994), Exp. No. XI, \'Ecole Polyt\'ech., Palaiseau, 1994. · Zbl 0881.58072
[97] 97 J. Newman and M. Z. Solomyak, Two-sided estimates on singular values for a class of integral operators on the semi-axis, Integral Equations Operator Theory 20 (1994), no. 3, 335-349. · Zbl 0817.47024
[98] 98 K. Naimark and M. Z. Solomyak, On the eigenvalue behaviour for a class of operators related to self-similar measures on \(\mathbfR^d, C\). R. Acad. Sci. Paris Ser. I Math. 319 (1994), no. 8, 837-842. · Zbl 0808.60038
[99] Sol_AaM_1994 M. Z. Solomyak, Piecewise-polynomial approximation of functions from Sobolev spaces, revisited, A tribute to Ilya Bakelman Discourses Math. Appl., vol. 3, Texas A & M Univ., College Station, TX, 1994, pp. 7-27.
[100] 100 M. Z. Solomyak, My recollections of Ilya Bakelman, A tribute to Ilya Bakelman Discourses Math. Appl., vol. 3, Texas A & M Univ., College Station, TX, 1994, pp. 28-30. · Zbl 0879.01031
[101] 101-a V. I. Arnold, M. Sh. Birman, M. Z. Solomyak, et al., Anatoli\vMoiseevich Vershik \rm (on the occasion of his sixtieth birthday), Uspekhi Mat. Nauk 49 (1994), no. 3, 195-204; English transl., Russian Math. Surveys 49 (1994), no. 3, 207-221.
[102] Sol_PLMS_1995 M. Z. Solomyak, Spectral problems related to the critical exponent in the Sobolev embedding theorem, Proc. London Math. Soc. (3) 71 (1995), no. 1, 53-75. · Zbl 0831.35116
[103] 102 M. Z. Solomyak and E. Verbitsky, On a spectral problem related to self-similar measures, Bull. London Math. Soc. 27 (1995), no. 3, 242-248. · Zbl 0823.34071
[104] 103 K. Naimark and M. Z. Solomyak, The eigenvalue behaviour for the boundary value problems related to self-similar measures on \(\mathbfR^d\), Math. Res. Lett. 2 (1995), no. 3, 279-298. · Zbl 0836.60016
[105] 104 M. Sh. Birman and M. Z. Solomyak, Tensor product of a finite number of spectral measures is always a spectral measure, Integral Equations Operator Theory 24 (1996), no. 2, 179-187. · Zbl 0844.47013
[106] BLSol_1997 M. Sh. Birman, A. Laptev, and M. Z. Solomyak, The negative discrete spectrum of the operator \((-\Delta )^l-\alpha V\) in \(L_2(\mathbfR^d)\) for \(d\) even and \(2l\geq d\), Ark. Mat. 35 (1997), no. 1, 87-126. · Zbl 0876.35079
[107] LeSol_1997 D. Levin and M. Z. Solomyak, The Rozenblum-Lieb-Cwikel inequality for Markov generators, J. Anal. Math. 71 (1997), 173-193. · Zbl 0910.47017
[108] 107 M. Z. Solomyak, On the negative discrete spectrum of the operator \(-\Delta_\mathcal N-\alpha V\) for a class of unbounded domains in \(R^d\), Partial Differential Equations and their Applications (Toronto, ON, 1995), CRM Proc. Lecture Notes, vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 283-296.
[109] RSol_1997 G. V. Rozenblum and M. Z. Solomyak, CLR-estimate revisited: Lieb’s approach with no path integrals. Journees ,,Equations aux Derivees Partielles” (Saint-Jean-de-Monts, 1997), Exp. No. XVI, Ecole Polytech., Palaiseau, 1997. · Zbl 1027.47020
[110] NSol_1997 K. Naimark and M. Z. Solomyak, Regular and pathological eigenvalue behavior for the equation \(-\lambda u''=Vu\) on the semiaxis, J. Funct. Anal. 151 (1997), no. 2, 504-530. · Zbl 0895.34063
[111] RSol_AA1997 G. V. Rozenblum and M. Z. Solomyak, The Cwikel-Lieb-Rozenblyum estimator for generators of positive semigroups and semigroups dominated by positive semigroups, Algebra i Analiz 9 (1997), no. 6, 214-236; English transl., St. Petersburg Math. J. 9 (1998), no. 6, 1195-1211.
[112] 111 G. V. Rozenblum and M. Z. Solomyak, On principal eigenvalues for indefinite problems in Euclidean space, Math. Nachr. 192 (1998), 205-223. · Zbl 0908.35085
[113] BLSol_1998 M. Sh. Birman, A. Laptev, and M. Z. Solomyak, On the eigenvalue behaviour for a class of differential operators on semiaxis, Math. Nachr. 195 (1998), 17-46. · Zbl 0916.34067
[114] 113 M. Z. Solomyak, On the discrete spectrum of a class of problems involving the Neumann Laplacian in unbounded domains, Voronezh Winter Math. Schools, Amer. Math. Soc. Transl. Ser. 2, vol. 184, Adv. Math. Sci., vol. 37, Amer. Math. Soc., Providence, RI, 1998, pp. 233-251. · Zbl 0906.35065
[115] 114 V. Buslaev, M. Z. Solomyak, and D. Yafaev, On the scientific work of Mikhail Shlemovich Birman, Differential Operators and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2, vol. 189, Adv. Math. Sci., vol. 41, Amer. Math. Soc., Providence, RI, 1999, pp. 1-15. · Zbl 1026.01503
[116] 115 G. V. Rozenblum and M. Z. Solomyak, On the number of negative eigenvalues for the two-dimensional magnetic Schr\"odinger operator, Differential Operators and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2, vol. 189, Adv. Math. Sci., vol. 41, Amer. Math. Soc., Providence, RI, 1999, pp. 205-217. · Zbl 0920.35101
[117] NSol_2000 K. Naimark and M. Z. Solomyak, Eigenvalue estimates for the weighted Laplacian on metric trees, Proc. London Math. Soc. (3) 80 (2000), no. 3, 690-724. · Zbl 1046.34092
[118] 117 M. Z. Solomyak, Estimates for the approximation numbers of the weighted Riemann-Liouville operator in the spaces \(L_p\), Complex Analysis, Operators, and Related Topics, Oper. Theory Adv. Appl., vol. 113, Birkh\"auser, Basel, 2000, pp. 371-383. · Zbl 0976.47028
[119] 117-a V. S. Buslaev, A. M. Vershik, M. Z. Solomyak et al., Mikhail Shl\"emovich Birman (on the occasion of his seventieth birthday), Uspekhi Mat. Nauk 55 (2000), no. 1, 204-207; English transl., Russian Math. Surveys 55 (2000), no. 1, 201-205. · Zbl 0958.01026
[120] 126 M. Z. Solomyak, A few words about Viktor Abramovich Zalgaller, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), 22-27; English transl., J. Math. Sci. (N.Y.) 119 (2004), no. 2, 138-140. · Zbl 1077.01533
[121] 118 M. Sh. Birman and M. Z. Solomyak, On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential, J. Anal. Math. 83 (2001), 337-391. · Zbl 1200.35196
[122] NSol_2001 K. Naimark and M. Z. Solomyak, Eigenvalue distribution of some fractal semi-elliptic differential operators: combinatorial approach, Integral Equations Operator Theory 40 (2001), no. 4, 495-506. · Zbl 0983.35096
[123] NSol_RJMP_2001 K. Naimark and M. Z. Solomyak, Geometry of Sobolev spaces on regular trees and the Hardy inequalities, Russ. J. Math. Phys. 8 (2001), no. 3, 322-335. · Zbl 1187.46028
[124] SoSol_2002 A. V. Sobolev and M. Z. Solomyak, Schr\"odinger operators on homogeneous metric trees: spectrum in gaps, Rev. Math. Phys. 14 (2002), no. 5, 421-467. · Zbl 1038.81023
[125] Sol_2002 M. Z. Solomyak, On eigenvalue estimates for the weighted Laplacian on metric graphs, Nonlinear Probl. Math. Phys. and Related Topics. I, Int. Math. Ser. (N. Y.), vol. 1, Kluwer/Plenum, New York, 2002, pp. 327-347. · Zbl 1045.35048
[126] 123 M. Z. Solomyak, On approximation of functions from Sobolev spaces on metric graphs, J. Approx. Theory 121 (2003), no. 2, 199-219. · Zbl 1027.46037
[127] Sol_2003 M. Z. Solomyak, Laplace and Schr\`“odinger operators on regular metric trees: the discrete spectrum case, Function Spaces, Differential Operators and Nonlinear Analysis (Teistungen, 2001), Birkh\'”auser, Basel, 2003, pp. 161-181.
[128] BSol2003_DOI M. Sh. Birman and M. Z. Solomyak, Double operator integrals in a Hilbert space, Integral Equations Operator Theory 47 (2003), no. 2, 131-168. · Zbl 1054.47030
[129] Sol_WRM_2004 M. Z. Solomyak, On a differential operator appearing in the theory of irreversible quantum graphs, Waves Random Media 14 (2004), no. 1, 173-185.
[130] 128 M. Z. Solomyak, On the spectrum of the Laplacian on regular metric trees, Waves Random Media 14 (2004), no. 1, 155-171.
[131] Sol_FAA_2004 M. Z. Solomyak, On the discrete spectrum of a family of differential operators, Funktional. Anal. i Prilozhen. 38 (2004), no. 3, 70-78; English transl., Funct. Anal. Appl. 38 (2004), no. 3, 217-223. · Zbl 1084.47038
[132] 130 V. S. Buslaev, M. Z. Solomyak, and D. R. Yafaev, Mikhail Shlemovich Birman (on the occasion of his \rm 75th birthday, Algebra i Analiz 16 (2004), no. 1, 5-14; English transl., St. Petersburg Math. J. 16 (2005), no. 1, 1-8. · Zbl 1061.01500
[133] 130-a V. I. Arnold, M. Sh. Birman, M. Z. Solomyak et al., Ol’ga Aleksandrovna Ladyzhenskaya, Uspekhi Mat. Nauk 59 (2004), no. 3, 151-152; English transl., Russian Math. Surveys 59 (2004), no. 3, 553-555. · Zbl 1062.01503
[134] EvSol-1 W. D. Evans and M. Z. Solomyak, Smilansky’s model of irreversible quantum graphs. I. The absolutely continuous spectrum, J. Phys. A 38 (2005), no. 21, 4611-4627. · Zbl 1064.81037
[135] EvSol-2 W. D. Evans and M. Z. Solomyak, Smilansky’s model of irreversible quantum graphs. II. The point spectrum, J. Phys. A 38 (2005), no. 35, 7661-7675. · Zbl 1083.81519
[136] NSol_2006 S. N. Naboko and M. Z. Solomyak, On the absolutely continuous spectrum in a model of an irreversible quantum graph, Proc. London Math. Soc. (3) 92 (2006), no. 1, 251-272. · Zbl 1098.81036
[137] Sol_2005 M. Z. Solomyak, A mathematical model for a noninvertible quantum graph, Algebra i Analiz 17 (2005), no. 5, 190-231; English transl., St. Petersburg Math. J. 17 (2006), no. 5, 835-864. · Zbl 1264.47051
[138] Sol_2006 M. Z. Solomyak, On the limiting behaviour of the spectra of a family of differential operators, J. Phys. A 39 (2006), no. 33, 10477-10489. · Zbl 1105.35064
[139] 136 U. Smilansky and M. Z. Solomyak, The quantum graph as a limit of a network of physical wires, Quantum Graphs and their Applications, Contemp. Math., vol. 415, Amer. Math. Soc., Providence, RI, 2006, pp. 283-291. · Zbl 1119.34026
[140] RSol_2007 G. V. Rozenblum and M. Z. Solomyak, On a family of differential operators with the coupling parameter in the boundary condition, J. Comput. Appl. Math. 208 (2007), no. 1, 57-71. · Zbl 1146.35064
[141] FrSol_RJMP_2008 L. Friedlander and M. Z. Solomyak, On the spectrum of narrow periodic waveguides, Russ. J. Math. Phys. 15 (2008), no. 2, 238-242. · Zbl 1180.35392
[142] 139 M. Z. Solomyak and T. A. Suslina, On the scientific work of M. Sh. Birman in 1998-2007, Spectral Theory of Differential Operators, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Adv. Math. Sci., vol. 62, Amer. Math. Soc., Providence, RI, 2008, pp. 1-15. · Zbl 1159.01011
[143] FrSol_2008 L. Friedlander and M. Z. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow infinite strip, Spectral Theory of Differential Operators, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Adv. Math. Sci., vol. 62, Amer. Math. Soc., Providence, RI, 2008, pp. 103-116. · Zbl 1170.35487
[144] RSol_2009 G. V. Rozenblum and M. Z. Solomyak, Counting Schr\"odinger boundstates: semiclassics and beyond, Sobolev Spaces in Mathematics. II, Int. Math. Ser. (N. Y.), vol. 9, Springer, New York, 2009, pp. 329-353. · Zbl 1172.35046
[145] 142 M. Z. Solomyak, Remarks on counting negative eigenvalues of the Schr\"odinger operator on regular metric trees, Linear and Complex Analysis, Amer. Math. Soc. Transl. Ser. 2, vol. 226, Adv. Math. Sci., vol. 63, Amer. Math. Soc., Providence, RI, 2009, pp. 165-178. · Zbl 1196.34037
[146] FrSol_2009 L. Friedlander and M. Z. Solomyak, On the spectrum of the Dirichlet Laplacian in a narrow strip, Israel J. Math. 170 (2009), 337-354. · Zbl 1173.35090
[147] RSol_Probl_2009 G. V. Rozenblum and M. Z. Solomyak, On the spectral estimates for the Schr\"odinger operator on \(\mathbb Z^d,\ d\ge 3\), Probl. Mat. Anal. 41 (2009), 107-126; English transl., J. Math. Sci. (N.Y.) 159 (2003), no. 2, 241-263. · Zbl 1207.35237
[148] 145 M. Z. Solomyak, T. A. Suslina, and D. R. Yafaev, Obituary: Mikhail Birman, 17.01.1928-2.07.2009, Math. Model. Nat. Phenom. 5 (2010), no. 4, 1-3. · Zbl 1219.01025
[149] RSol_2010 G. V. Rozenblum and M. Z. Solomyak, On spectral estimates for Schr\"odinger-type operators: the case of small local dimension, Funktsional. Anal. i Prilozhen. 44 (2010), no. 4, 21-33; English transl., Funct. Anal. Appl. 44 (2010), no. 4, 259-269. · Zbl 1272.47055
[150] 147a V. M. Babich, V. S. Buslaev, M. Z. Solomyak et al., Mikhail Shl\"emovich Birman, Uspekhi Mat. Nauk 65 (2010), no. 3, 185-190; English transl., Russian Math. Surveys 65 (2010), no. 3, 569-575. · Zbl 1219.01025
[151] BSol_book_2ed M. Sh. Birman and M. Z. Solomyak, Spectral theory of selfadjoint operators in Hilbert space, 2nd ed., revised and augmented, Lan\textprime , St. Petersburg, 2010.
[152] RSol_2011 G. V. Rozenblum and M. Z. Solomyak, Spectral estimates for Schr\"odinger operators with sparse potentials on graphs, Probl. Mat. Anal. 57 (2011), 151-164; English transl., J. Math. Sci. (N.Y.) 176 (2011), no. 3, 458-474. · Zbl 1290.35284
[153] 149 M. Z. Solomyak, T. A. Suslina, and D. R. Yafaev, On the mathematical works of M. Sh. Birman, Algebra i Analiz 23 (2011), no. 1, 5-60; English transl., St. Petersburg Math. J. 23 (2012), no. 1, 1-38. · Zbl 1236.01029
[154] LSol_2012 A. Laptev and M. Z. Solomyak, On the negative spectrum of the two-dimensional Schr\"odinger operator with radial potential, Comm. Math. Phys. 314 (2012), no. 1, 229-241. · Zbl 1253.35037
[155] Sol_2013 M. Z. Solomyak, On a class of spectral problems on the half-line and their applications to multi-dimensional problems, J. Spectr. Theory 3 (2013), no. 2, 215-235. · Zbl 1270.34197
[156] LSol_2013 A. Laptev and M. Z. Solomyak, On spectral estimates for two-dimensional Schr\"odinger operators, J. Spectr. Theory 3 (2013), no. 4, 505-515. · Zbl 1294.35048
[157] RSol_2013 G. V. Rozenblum and M. Z. Solomyak, On spectral estimates for the Schr\"odinger operators in global dimension \(2\), Algebra i Analiz 25 (2013), no. 3, 185-199; English transl., St. Petersburg Math. J. 25 (2014), no. 3, 495-505. \pagebreak
[158] 154 V. Moroz, M. Z. Solomyak, and A. Truman, Vitali Liskevich, Proc. Edinb. Math. Soc. (2) 57 (2014), no. 2, 573-574.
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