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A (bounded linear) Hilbert space operator      T T       is called  quasitriangular  if there exists an increasing sequence      { P n } n = 0 ∞  \{ {P_n}\} _{n = 0}^\infty       of finite-rank orthogonal projections, converging strongly to 1, such that      ‖ ( 1 −  P n ) T P n ‖ →  0 ( n →  ∞  ) \left \| {(1 - {P_n})T{P_n}} \right \| \to 0\,(n \to \infty )      . This definition, due to P. R. Halmos, plays a very important role in operator theory. The core of this article is a concrete answer to the following problem: Suppose      T T       is a quasitriangular operator and      Γ  = { λ  j } j = 1 ∞  \Gamma = \{ {\lambda _j}\} _{j = 1}^\infty       is a sequence of complex numbers. Find necessary and sufficient conditions for the existence of a compact operator      K K       (of arbitrarily small norm) so that      T −  K T - K       is triangular with respect to some orthonormal basis, and the sequence of diagonal entries of      T −  K T - K       coincides with      Γ  \Gamma      . For instance, if no restrictions are put on the norm of      K K      , then      T T       and      Γ  \Gamma       must be related as follows: (a) if      λ  0 {\lambda _0}       is a limit point of      Γ  \Gamma       and      λ  0 −  T {\lambda _0} - T       is semi-Fredholm, then      ind ( λ  0 −  T ) &gt; 0 {\operatorname {ind}}({\lambda _0} - T) &gt; 0      ; and (b) if      Ω  \Omega       is an open set intersecting the Weyl spectrum of      T T      , whose boundary does not intersect this set, then      { j : λ  j ∈  Ω  } \{ j:{\lambda _j} \in \Omega \}       is a denumerable set of indices. Particularly important is the case when      Γ  = { 0 , 0 , 0 , …  } \Gamma = \{ 0,0,0, \ldots \}      . The following are equivalent for an operator      T T      : (1) there is an integral sequence      { P n } n = 0 ∞  \{ {P_n}\} _{n = 0}^\infty       of orthogonal projections, with rank      P n = n {P_n} = n       for all      n n      , converging strongly to 1, such that      ‖ ( 1 −  P n ) T P n + 1 ‖ →  0 ( n →  ∞  ) \left \| {(1 - {P_n})T{P_{n + 1}}} \right \| \to 0\,(n \to \infty )      ; (2) from some compact      K , T −  K K,\,T - K       is triangular, with diagonal entries equal to 0; (3)      T T       is quasitriangular, and the Weyl spectrum of      T T       is connected and contains the origin. The family      ( StrQT ) −  1 {({\text {StrQT}})_{ - 1}}       of all operators satisfying (1) (and hence (2) and (3)) is a (norm) closed subset of the algebra of all operators; moreover,      ( StrQT ) −  1 {({\text {StrQT}})_{ - 1}}       is invariant under similarity and compact perturbations and behaves in many senses as an analog of Halmos’s class of quasitriangular operators, or an analog of the class of extended quasitriangular operators      ( StrQT ) −  1 {({\text {StrQT}})_{ - 1}}      , introduced by the author in a previous article. If      { P n } n = 0 ∞  \{ {P_n}\} _{n = 0}^\infty       is as in (1), but condition      ‖ ( 1 −  P n ) T P n + 1 ‖ →  0 ( n →  ∞  ) \left \| {(1 - {P_n})T{P_{n + 1}}} \right \| \to 0\,(n \to \infty )       is replaced by (1’)      ‖ ( 1 −  P n k ) T P n k + 1 ‖ →  0 ( k →  ∞  ) \left \| {(1 - {P_{{n_k}}})T{P_{{n_k} + 1}}} \right \| \to 0\,(k \to \infty )       for some subsequence      { n k } k = 1 ∞  \{ {n_k}\} _{k = 1}^\infty      , then (1’) is equivalent to (3’),      T T       is quasitriangular, and its Weyl spectrum contains the origin. The family      ( QT ) −  1 {({\text {QT}})_{ - 1}}       of all operators satisfying (1’) (and hence (3’)) is also a closed subset, invariant under similarity and compact perturbations, and provides a different analog to Halmos’s class of quasitriangular operators. Both classes have “     m m      -versions” (     ( StrQT ) −  m {({\text {StrQT}})_{ - m}}       and, respectively,      ( QT ) −  m {({\text {QT}})_{ - m}}      ,      m = 1 , 2 , 3 , …  m = 1,2,3, \ldots      ) with similar properties. (     ( StrQT ) −  m {({\text {StrQT}})_{ - m}}       is the class naturally associated with triangular operators      A A       such that the main diagonal and the first      ( m −  1 ) (m - 1)       superdiagonals are identically zero, etc.) The article also includes some applications of the main result to certain nest algebras “generated by orthonormal bases.”

The diagonal entries in the formula “quasitriangular - compact = triangular” and restrictions of quasitriangularity