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
)
>
0
{\operatorname {ind}}({\lambda _0} - T) > 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.”