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$(S+N)$-triangular operators: spectral properties and important examples

We introduce a notion of $(S+N)$-triangular operators in the Hilbert space using some basic ideas from triangular representation theory. Our notion generalizes the well-known notion of the spectral operators so that many properties of the $(S+N)$-triangular operators coincide with those of spectral operators. At the same time we show that wide classes of operators are $(S+N)$-triangular.